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On some problems of interpolation and approximation theory
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Full Text

ON SOME PROBLEMS OF INTERPOLATION
AND APPROXIMATION THEORY

By

JOHN BURKETT

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1992

UWVaSITY OF FLORIDA LiNWR-0E

To my Parents

Frederick and Catherine Burkett

ACKNOWLEDGMENTS

The author wishes to express sincere thanks to Dr. Arun Kumar Varma for his many hours of discussions and counseling in the field of approximation theory. His contagious enthusiasm for the subject and his desire to share this enthusiasm made this dissertation possible. The author is grateful for the generous advice and comments made by the members of the Ph.D. committee. They are Dr. Edwards, Dr. Mair, Dr. Popov, and Dr. Sheppard, in addition to Dr. Varma. Special thanks belong to Dr. Szabodos for his kindness and suggestions during a visit to the University of Florida. Finally, the author thanks Dr. Drake and the Department of Mathematics for their support.

iii

ACKNOWLEDGMENTS .......................

ABSTRACT . . . . . . . . . . . . . . . . . . .

CHAPTERS

One INTRODUCTION .............

TWO

Polynomial Approximation ....... Lagrange and Hermite-Fej~r Interpolation Birkhoff Interpolation ........ Lacunary Spline Interpolation ....... Markov Type Inequalities .......

EXPLICIT REPRESENTATION OF A (0;2) PROCE

Introduction and Main Results .......
Preliminaries ..... ..............
Proof of Theorem 2.1 .........

Three CONVERGENCE RESULTS FOR A (0;2) PROCESS

Introduction and Main Results .......
Preliminaries ..... ..............
Lemmas . . . . . . . . . . . . . . . .
Proof of Convergence Results .....

Four LACUNARY INTERPOLATION BY SPLINES . . .

Introduction and Main Results .......
Proof of Theorems .... ............

Five EXTREMAL PROPERTIES FOR THE DERIVATIVES
OF ALGEBRAIC POLYNOMIALS .......

Main Results .............
Lemmas . . . . . . . . . . . . . . . .
Proof of Theorems .. ..............

Six SUMMARY AND CONCLUSIONS ............

REFERENCES ..................

Page

* . . . iii

* . . . 19 SS . . . 24 .... . 24
.... . 27
.... . 28

.... . 33

.... . 33
.... . 34
.... . 37
.... . 93

.... . 95

.... . 95
.... . 99

* . . . 110

* . . . 110 . .. . 112 . .. . 122 . .. . 131

* . . . 134

BIOGRAPHICAL SKETCH ... ........

. . . . . . . . . 138

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
ON THE PROBLEMS OF INTERPOLATION AND APPROXIMATION THEORY

By

JOHN BURKETT

December 1992

Chairman: Dr. Arun K. Varma Major Department: Mathematics

We start by presenting some well known results

concerning polynomial approximation, Birkhoff interpolation, lacunary spline interpolation, and Markov type inequalities. These results provide a historical motivation for the problems considered in later chapters.

After this introduction, we give the explicit

representation and convergence properties for a Birkhoff interpolation process. We find the unique algebraic polynomial that takes on function values for a specific set of knots. In addition, the second derivative of this polynomial takes the value zero on another set of prescribed knots. Next, we look at how well these polynomials approximate a given function. It is shown that the polynomials converge uniformly for any continuous function on the closed interval. In fact, we present a pointwise

estimate which provides a discrete, interpolatory proof of the Teljakovskii Theorem of Approximation.

Next, motivated by earlier results of Meir and Sharma, we consider lacunary spline interpolation in the (0,1,3) and (0,1,2,4) cases. For (0,1,3) interpolation, function values and third derivative values are prescribed at the joints, while function values and first derivative values are prescribed at the midpoints of the joints. Similarly, we consider the problem of (0,1,2,4) interpolation. One such spline turns out to be local in character. Specifically, it is determined by the solution of a diagonal matrix.

Motivated by an open problem of P. Turin, Rabman

studied the extremal properties of polynomials under curved majorants in the uniform norm. We discuss similar results in the LP norm. We present theorems for both the circular and parabolic majorants.

CHAPTER ONE
INTRODUCTION

Polynomial Approximation

In 1885, Weierstrass showed that an arbitrary

continuous function on a compact interval can be uniformly approximated by a sequence of polynomials. We more precisely state this as follows.

Let f e C[-1,1]. Here C[-1,1] denotes the class of

functions continuous on [-1,1]. For any E > 0, there exists a polynomial P(x) such that jif(x) - P(x)jI < E. Here, we denote the usual sup norm,

-l = x - If(x) l

In 1909, Dunham Jackson chose, as a thesis topic, to investigate the degree of approximation with which a given continuous function can be represented by a polynomial of given degree [21]. This gave rise to the Jackson Theorem.

Before presenting the theorem, we define the concept of best approximation for a given function. Let f E C[-1,1]. Then

inf
(i~~i) E=() =P'GIT, If - P, I

where n. represents the class of algebraic polynomials of degree - n.

Theorem 1.1 (Jackson). Let f e C[-1,1]. There exists a positive constant A such that (1.1.2) E,(f) Aw(f,-!) for n = 1, 2,
n

where A is independent of f.

Note: w(f,6) represents the usual modulus of continuity defined by
(f,8) Isu,'f If (x) - f (Y) I

A more rapid decrease to zero for En(f) is possible if we assume more smoothness for f. Dunham Jackson proved the following result on differentiable functions.

Let Cr[-1,1] denote the class of functions f such that
fcr) e C[-1,1].

Theorem 1.2 (Jackson). If f E Cr[-l1,1], then (1.1.3) E,(f) Ar (1)r w (f(r), 1) for n = 1, 2, n n

where A is a constant independent of f.

In 1951, A.F. Timan obtained the following improvement of the Jackson Theorem. Theorem 1.3 (Timan). Let f E C[-1,1]. There exists a positive constant B and a polynomial P, of degree n such that

(1.1.4) jf(x) - P,,(x) I !g B Wi f, 1L-7i) + CO(.t. J2)1

for -1 g x g 1 and n = 1, 2, ... where B is independent of f.

Timan's theorem gives a pointwise estimate for If(x) P.(x) l. Notice that as Ixi approaches 1, the order of convergence near the ends is better than at the middle of [-1,1].

We now state a needed definition. Let Lip a represent the class of functions f in C[-1,1] such that

(1.1.5) If(x) -f(y) I g Mlx-yla for all x and y in [-1, 1]

where M is some fixed constant.

We now state a converse to the Timan Theorem proved by V.K. Dzjadyk in 1956.

Theorem 1.4 (Dzjadyk). Let f E C[-1,1] and 0 < a < 1. There exists a constant B and a polynomial P. of degree n such that

(1.1.6) If(x) - P"(x) I [!g B + ) for -i x r 1

and n = 1, 2, ... if and only if w(f,h) Ch' for some constant C.

4

A further improvement of the Jackson and Timan Theorems was made by Teljakovskii in 1966. Here, it was shown that the estimate of If(x) - P.(x) I can be made exact at the endpoints of the interval.

Theorem 1.5 (Teljakovskii). Let f E C [-1, 1]. There exists a positive constant D and a polynomial P. of degree n such that

(1.1.7) If(x)-P,(x)l : DGaf,J ) for -1 ! x : 1 and

n = 1, 2, ... where D is independent of f.

Modifying the Jackson operator, Ron DeVore strengthened the above theorem as follows [10].

For a function f(x) bounded on [-1,1], we define the second modulus of smoothness by (1.1.8) ( sup If(x) -2f (x+h) +f(x+2h) I"
(1.1 8) "(f, 8) = _l:sx,x+2h:5l, Ihj58

Theorem 1.6 (DeVore). Let f E C[-1,1I]. There exists a positive constant A and a polynomial P. of degree n such that

(1.1.9) If(x)-P2(x) I :A02(f, inX) for-1 r x 1 and

n = 1, 2, ... where A is independent of f.

There are a number of proofs for the Teljakovskii and

DeVore Theorems. The initial proofs involved convolution of the approximated function with the Jackson kernel. This

requires the function to be known almost everywhere. In 1979, Mills and Varma obtained a discrete, weakly interpolatory proof of the Teljakovskii Theorem [24]. In 1989, Varma and Yu obtained such a proof for the DeVore Theorem [48]. These proofs require the function values to be known at only a discrete number of points.

This concludes our brief discussion on polynomial approximation. The reader may obtain other important contributions from the book of Timan [39].

Lagrange and Hermite - Fejir Interpolation

Let X denote an infinite triangular matrix with all entries in [-1,1]

x0, 0
(1.2.1) X X0, X1,
Xo, n X1,n "" Xn,n

We denote by Ln[f,x;X], the Lagrange polynomial of degree < n which interpolates f(x) at the nodes xk,. for k = 0, 1, ..., n. Then

n
(1.2.2) L[f,x;X] = E f (xk,) lk,,.(x) k-0

where

'k.n (x) = _ _ _ _x) n
k'kni(k), nX(x) = R (x-xk, n)
(x-xk,,=) Can (xkn) k-0

For a time, it was thought that for some matrix X, the Lagrange interpolating polynomials converge uniformly to any given continuous function on [-1,1]. The hopes for this idea vanished when Bernstein and Faber simultaneously discovered in 1914 that for any triangular system of interpolation points, we can construct a continuous function for which the Lagrange interpolatory process carried out on these points cannot converge uniformly to this function.

In 1916, L. Fejr showed that if instead of Lagrange interpolation we consider Hermite-Fej~r interpolation the situation changes [17]. The Hermite-Fej6r polynomials H,[f,x;X] are of degree g 2n + 1 and uniquely determined by the conditions

(1.2.3) HIf,x,;X] = f(xkl), H[f, xkf;X] = 0 for k = 0, 1, ..., n. Fej~r showed that for particular matrices X,. as in (1.2.1), the Hermite-Fej~r interpolating polynomials converge uniformly to any given function f E C[-1,1]. For example, choosing the knots to be the zeros of the Tchebycheff polynomial T.(x) = cos ne, x = cose guarantees convergence for the entire class of continuous functions on the closed interval [-1,1].

Birkhoff Interpolation

In problems of Hermite type interpolation, function

values and consecutive derivative values are prescribed at

given points. In 1906, G.D. Birkhoff considered those interpolation problems in which the consecutive derivative requirement can be dropped [6]. This more general kind of interpolation is now referred to as Birkhoff (or lacunary) interpolation.

The problems of Birkhoff interpolation differ greatly from Lagrange and Hermite interpolation. For example, Lagrange and Hermite interpolation problems are always uniquely solvable for a given set of knots, but a given problem in Birkhoff interpolation may not have a unique solution.

More precisely, given n + 1 integer pairs (i,k)

corresponding to n + 1 real numbers Yi,k and m distinct real numbers xi, i = 1, 2, ..., m : n + 1, a given problem of polynomial interpolation is to satisfy the n + 1 equations

(1.3.1) p W) (xk ) =

with a polynomial P. of degree at most n. (We use the convention P,(O0(x) = P.(x).) For each i, the orders k of the derivatives in (1.3.1) form a sequence k = 0, 1, ..., ki. If one or more of the sequences is broken, we have Birkhoff interpolation.

A number of different cases in Birkhoff interpolation have been studied. In its first general treatment, Turdn and associates studied (0,2) interpolation where the knots

are the zeros of the integral of the Legendre polynomial [3][4][34]. It was found that these interpolating polynomials exist uniquely only when the number of knots used is even. We state this result as a theorem. Define

(1.3.2) 9n (x) = (1-x2) P", (x) where Pnl(x) is the Legendre polynomial of degree n - 1 normalized by P.1(1) = 1. An equivalent definition of nn(X) is

x
(1.3.3) =x)=-n(n-l)fxP ,td:
-1

Theorem 1.7 (Turin and Suranyi). Given arbitrary real

values (al,n, a2,n, ... , a.,.) and (bl,n, b2,n, ..., b,,,) where n is an even positive integer, there exists a unique real algebraic polynomial R.(x) of degree < 2n - 1 such that

(1.3.4) R, (x,) = a,,, and R," (xi,) =b1,n

for i = 1, 2, ..., n where

= X,n < X2, < ... < Xn,n =1

are the zeros of (1 - x2)P'_(x).

Later, Varma and Prasdd proved the following [47].

Theorem 1.8 (Varma and Prasid). Given arbitrary real values

(c1,, c2,n, . .. , c.,n) and (d2,., d3,., ... , d.-,,.) where n is

an even positive integer, there exists a unique real algebraic polynomial Q.(x) of degree < 2n - 3 such that

(1.3.5) Qn(xi,,) = ci,, for i = 1,2, ...,n and

Qi (xi,2) = di,, for i = 2, 3, ..., n - 1 where

-1 = xl,n < x2,n < ... < X,. = 1 are the zeros of (1 - x2) P,-2(x).

After answering the questions of existence and uniqueness, it is natural to address the problem of convergence. Turn and Baldzs followed Theorem 1.6 with a result on convergence which was subsequently improved by Freud (18].

Theorem 1.9' (Turdn and Bal~zs, improved by Freud). Let f E C[-1,1] such that W2(f,h) = h e(h) where E(h) - 0 as h - 0.

If IPi,nj < S. where n-1 6. - 0, then

(1.3.6) Rn (f,x) - f(x) uniformly for -i < x < 1.

Here, R,(f,x) is given by

10

n n
(1.3.7) R,(f,x) = E f(xk,n) rk,(x) + E k,nPkn(x) k-i k-1

where the explicit forms of rk,.(x) and pk,.(x) are given by Baldzs and Turin [3].

Other cases of (0,2) interpolation have also been studied. For example, Varma studied the convergence properties of the (0,2) interpolating polynomials where the knots are the zeros of Tchebycheff polynomials of the first kind [41]. Here, it has been shown by SurAnyi and Turin, the polynomials exist uniquely for an even number of knots [34].

Theorem 1.10 (Varma). Let f e C1 [-1,1] and let f'E Lip a, > 1
2

If

(1.3.8) ai= rn for i = 2, 3, ... , n + 1

where xi,. are the zeros of T,(x) = cos nO, x = cosO and

(1.3.9) n 0.
n-00 0

Then Sn(f,x) converges uniformly to f(x) in [-1,1] where

n+1 n+1
(1.3.10) Sn(f,x) = E f(xi,n) ui,n(x)+E 6i,n vi,(x).
i=2 1-2

Here uï¿½,,(x) and vi,,(x) are given by Varma [41].

Saxena and Sharma extended (0,2) interpolation to the case of (0,1,3) interpolation [33]. They considered the problems of existence and uniqueness of polynomials which interpolate prescribed function values, first derivative values, and third derivative values at a given set of points. They also obtained convergence results analogous to Theorem 1.9.

In 1989, Akhlaghi, Chak, and Sharma addressed the

problem of (0,3) interpolation based on the zeros of u.(x) [1]. They found the (0,3) interpolating polynomials to exist uniquely for every n 4. In addition, explicit forms were found for the fundamental polynomials though complicated in nature.

Subsequently, Szabodos and Varma [36] found simpler explicit forms for this (0,3) interpolation and, consequently, were able to obtain the following convergence result.

Theorem 1.11 (Szabodos and Varma). Let f E C[-1,1]. Then

(1.3.11) ilf(x) - Rn(f,x) 0 (43(ft logl/3 n))

where 3(f,h) is the modulus of smoothness of order 3 of f(x). Here

n
(1.3.12) R,(f,x) = E f(xj,) rj,(x) j=1

where rj,.(x) are the fundamental polynomials of the first kind.

Notice that these (0,3) interpolating polynomials

converge uniformly for a wider class of functions than in the (0,2) interpolation theorems we presented. In fact, there is an open problem of Turin to find the "most stable" (0,2) interpolation in the following sense [40]. Problem XXXI.

Given a matrix X as in (1.2.1) such that each row

contains knots where the (0,2) interpolating polynomials exist, find the matrix that will minimize (1.3.13) max .

where rk,,(x) are the fundamental polynomials of the first kind.

Since rk,. (xk,n) = 1, we cannot hope to do better than

max s
(1.3.14) -x rk( xI = (1) for some matrix x.

Although (1.3.14) has not been obtained in the

strictest sense, such a result has been recently obtained by the use of two different sets of knots. This type of

interpolation has been referred to as P l Type interpolation.

Akhlaghi and Sharma have studied (0,2) interpolation on two different sets of knots, namely the zeros of (1 - x2) P _1(x) and P._1(x) [2]. They established that these interpolating polynomials exist uniquely for n even or odd. In addition, some results on explicit forms were obtained which we will not state here. We do, however, present the following.

Theorem 1.12 (Akhlaghi and Sharma). Given arbitrary real values (al,n, a2,n, ... , an,n) and (bl,n, b2,n, ... , bn_1,), there exist unique real algebraic polynomials S,(x) and T,(x) of degree 2n - 2 each such that

(1.3.15) S.(xi,.) = aj1. for i = 1, 2, ... , n and

S,,1(yi,,) =bi,, for i = 1, 2, ... , n - 1 and

(1.3.16) Tny,)=bi,,n for i 1 , 2, n. , - 1 and

=ai,, for i = 1, 2, ... , n. Here, xï¿½,. are the zeros of (1 - x2) Pl_1(x) and y1, are the zeros of Pn_1(x).

Subsequently, Szabodos and Varma proved uniqueness and existence for a modified (0,2) process very similar to the one described in (1.3.15). Their modified (0,2) process differs in that it also prescribes first derivative values

at the endpoints ï¿½ 1. Before presenting the convergence results, we define
n
(1.3.17) Ra(f,x) = E f(xk,n) rkfl(x) k=

and

n
R,(f,x) = f (xk,n) rk,f (X) (1.3.18) k-1
+ f(i) ol,n(X) + f(-l)a2,n(X)

We refer to the paper of Szabodos and Varma (35] for explicit forms. The following are their convergence theorems.

Theorem 1.13 (Szabodos and Varma). Let f E C[-1,1]. Then

(1.3.19) if (x) -R,(f,x) 1 = OH.f,2/liI)) f or -1 :5 x ! 1

and n = 1, 2,

Theorem 1.1,4 (Szabodos and Varma). Let f be a function such that f' e C[-1,1]. Then (1.3.20) If(x) - R,,F,xI = (1 2 T, W (f,) (n2 k=1, k

for -1 < x g 1 and n = 1, 2, .... A direct consequence of Theorem 1.12 is that (1.3.14) holds. This resolves Turdn's Problem XXXI in a slightly different context.

For more results on Birkhoff interpolation, we refer the reader to the book of Lorentz [223.

Lacunary Spline Interpolation

In the 1970s, 1980s, and 1990s, several papers appeared in which (0,2), (0,3), and (0,1,3) interpolation problems were solved using polynomial splines and piecewise polynomials. We can classify many of these results into three groups.

Before proceeding, we define by S(' the class of splines S(x) such that

(1.4.1) i) S(x) G C1[o,1]

ii) S(x) is a polynomial of degree q in [xi,xi+1], i = 0, 1, ..., n-i where
0 = x0 < x1 < ... < xn-1 < Xn = i

In this first group, the data to be interpolated are prescribed at the joints of a spline as well as at the endpoints of the interval. A. Meir and A. Sharma [23] were the first ones to consider the case of (0,2) interpolation on equidistant knots. They have shown that for arbitrary lacunary data {yi} 0 and {Yi)0, there exists a unique (up to the boundary conditions) quintic spline S.(x) E C'[0,1] with joints at - (i = 0, 1, ..., n) such that
n

SH i
,,( ) = yi, S n,--) =yiv (n odd). The boundary conditions are n n

Sn"(0) = /and S l~I) = yn"'. Moreover, if the given values

[yi], {y i], and {yt", yn .} are the values and the second and third derivative values, respectively, of a function f satisfying f e C4(0,1], Meir and Sharma proved the following convergence theorem.

Theorem 1.15 (Meir and Sharma). For the unique quintic spline S,(x) that interpolates (0,2) data as discussed, we have

(1.4.2) - _f(Z)I g 75n -3 Ca (f(4),1) + 8fnr-4 If(4) 1
n

for r = 0,1,2,3.

Subsequently, S. Demko pointed out that because of the ill-poised nature of the interpolant defined in the preceding, for a given function f e C6[0,1], the error If - SJ where S, interpolates f (as described above), is not of optimal order as a function of mesh length [9]. He further gave justification to this claim. On the other hand, S. Demko also pointed out that the situation changes if, instead of considering (0,2) interpolation by splines, we consider the (0,3) case based on equidistant knots. Consider arbitrary lacunary data fy1j}.O, {y"1o, and

{Y.", yn"}. There exists a unique quintic spline S.(x) E C3[0,1] with joints at - (i = 0, 1, ..., n) such that
n

y1, S"' yj"', S," (0) = y, and S,'(1) = y". The
n n

system of equations that uniquely determines S,(x) turns out to be tridiagonal dominant, and consequently, the rate of convergence is of the same order as that of best approximation by quintic C3 splines, provided the interpolated data corresponds to the function approximated.

fy f(x,) , yfl = f"III(Xi) , y0I = .f"1(x") , y- f"l(X") , f r C3 [0,l]1

The second group of results deals with special

piecewise polynomial methods for solving (0,2), (0,2,3), and (0,2,4) problems. We refer to the work of Fawzy [13] [14] (15]. Later, Fawzy and Schumaker [16] defined construction methods for solving the general lacunary interpolation problem.

On the positive side, these methods are shown to deliver the optimal order of approximation while being relatively easy to construct. One possible defect remarked by them is that their proposed methods produce only piecewise polynomials. We refer here to remark 1 on page 424(16]. Here, one should note that the data are prescribed at the knots only.

For the third group of results dealing with lacunary interpolation by splines, we refer to the papers of A.K. Varma (42] [43], J. Prasad and A.K. Varma [28], Gary

18

Howell and A.K. Varma [20]. Here, we allow certain data to be prescribed at the midpoints of the joints, in addition to at the joints of the spline.

Howell and Varma [20] obtained deficient quartic

splines of the class C2[0,1] which interpolate lacunary data (function values at the midpoints of the joints, second derivative values at the joints, and function values at the endpoints of the interval).

They obtained the following convergence theorem for these splines.

Theorem 1.16 (Howell and Varma). Let f E C'[0,1]. Then, for the unique quartic spline Sn(x) associated with f and satisfying the above conditions, we have

(1.4.3) ISr) - f (r)(x) I C' 1h 1 (f(1 ,h) , r = 0,1 and

1 = 2,3,4, and

(1.4.4) IS.r) (x) - f ()(x) BF h5--r max If( (x)1, r = 0,1

and 1 = 5, where h is the mesh length.

The splines in this group are determined by tridiagonal dominant systems. In fact, we now present a case where the spline is actually determined by a diagonal matrix. Pras~d and Varma obtained the first such case. They prescribed function values at the joints and midpoints of the joints, third derivative values at the midpoints of the joints, and

first derivative values at the endpoints of the interval. They proved the following convergence theorem. Theorem 1.17 (Pras&d and Varma). Let f e C1[0,1]. Then, for the unique quintic spline S,(x) associated with f and satisfying the above conditions, we have

(1.4.5) Is(z) (x) - fr (x) E 1 h ((f h) , r = 0,1, 2

and 1 = 3, 4, 5, and

(1.4.6) ISIr) (x) - f(r) (x) h' 6 - maX If ()

r = 0,1,2 where h is the mesh length.

This concludes our discussion on lacunary spline interpolation.

Markov Type Inequalities

In 1889, A.A. Markov proved the following.

Theorem 1.18 (Markov). If P,(x) is a real algebraic polynomial of degree n such that IP,(x) I 1 on the interval -1 < x 1, then we have

-max, lP (x)l [ n2
(1.5.1) -l x

Later, A. Zygmund [49] proved

Theorem 1.19 (Zygmund). If f is a trigonometric polynomial of order n and p k 1, then (1.5.2) { If') d n { f If() IPd

Hill, Szeg6 and Tamerkin [19] extended this type of

inequality to algebraic polynomials on the interval [-1,1] in the form

(1.5.3 ( IP,(M) IP dx f IP.(X) 1Pdx)

where p > 1 and A is independent of n and P,(x). They noted that the problem of obtaining the best constant in (1.5.3) is extremely difficult. Later, B.D. Bojanov [7] proved the following extension of the Markov Inequality. Theorem 1.20 (Bojanov). Let 1 < p < . Then for every real algebraic polynomial of degree n, we have

(1.5.4) I Px :g f IT,(x) l P max IPn(x) I
"- -1 x ~l

where Tn(x) = cos nO, x = cos 0.

The following problem was raised by P. Turin at a

conference held in Varna, Bulgaria (1970). Let *(x) k 0 for

-1 < x < 1 and consider the class Pn,* of all polynomials

n
Pn(x) = Z akxk of degree at most n such that IP (x) I K0(x)
k=0

for -i < x 1 1. How large can max k) (x) become if P,(x) is an arbitrary polynomial belonging to P,, ? Important contributions to the problem of Turin have been made by Rahman and his associates. The results we state involve the circular ((x) = ) and the parabolic majorants (O(x) = 1 - x2).

Theorem 1.21 (Rahman). If P,(x) is an algebraic polynomial of degree n such that IP,(x) I < - for -1 < x < 1, then

max JI(x) I 2 (n-1)

Equality if and only if

P"(x) = (1 x2) u,_2(x), u_2(x) = sin(n-l) x cosO.
sinO

Theorem 1.22 (Rahman and Watt). For given n k 3, let (1.5.6) zr zn = cos(- r = 0, 1, ..., n-2. If P(x) = (1 - x2)q(x) is a polynomial of degree at most n such that lq(A1) 1 < 1 for r = 0, 1, ..., n-2, then (1.5.7) jp(k) : g 1T(k)(1)I for k = 3, 4,

where zr(x) = (l-x2) T_2(x), T,-2(x) = cos (n-2)8, x = cosO.

Further, if P(x) is real for real x, then

(1.5.8) IP(k)(x+iy)I < -r k) (I+/y) I for (x,y) E [-1,1] x R

and k = 3, 4,....

For other interesting results, we refer to the works of Rahman and associates [25] [26] [27] [29] [30] [31].

A natural extension of these ideas is to investigate

similar problems in the LP norm. We shall state two such results obtained by Varma and associates in the L2 norm. Theorem 1.23 (Varma). Let Pn+1(x) by any real algebraic

polynomial of degree at most n + 1 such that IPn 1 (x) I < Vr-x for -1 < x < 1, then

1 1 1 1
( (x) ]2 (L-2)2 dx f [fW (x)]2 (i-X2)2 dx
-1 -1

for j = 1, 2, 3 where f.(x) = (l-x2) u"_ (x), u"_,(x) = sin nO ' i -isine

x = cosO. Equality if and only if P.+1(x) = + fo(x). Theorem 1.24 (Varma, Mills, and Smith). Let P, 2(x) be any real algebraic polynomial of degree at most n + 2 such that IP.2(x) I < l-x2 for -1 < x < 1, then

1 1
(1.5.10) f [P,,,+ 2(X) ] dX f [ftI(X)1]2 dX
-1 -1

23

where f1(x) = (1 - x2)T,(x), T,(x) = cos nO, x = cosO. Equality if and only if P,+2(x) = ï¿½ f1(x).

For other results in the L2 norm, we refer to the work of Varma and associates [44] [45] [46].

CHAPTER TWO
EXPLICIT REPRESENTATION OF A (0;2) PROCESS

Introduction and Main Results

Define
(2.1.1) -1 = to,n < t,n < . , tn,n = 1 to be the zeros of (1 - x2)Pn_1(x), and

(2.1.2) -1 < x2,. < x3,n , ... < xnl,n < to be the zeros of P.-' (x). Here, P,_1(x) denotes the Legendre polynomial of degree n - 1 with normalization

(2.1.3) P.-1 (1) = 1.

The following theorem is a direct result of Lemma2 in a paper by Akhlaghi and Sharma [2]. Theorem 2A

Given arbitary values (ao,n, aj,n, ..., a,,) and (b2,n, b3,., ..., bn_1,), there exists a unique real algebraic polynomial Rn(x) of degree 2n - 2, such that (2.1.4) R. (tj,,) = aj,n for j = 1, 2, ..., n - 1, (2.1.5) R. (-1) = a.,. , Ro(1) =an,n, (2.1.6) R,'(xj,=) =bjoa for j =2, 3, ..., n- 1.

25
We note that the above theorem places no restriction on n being even or odd. This is in contrast to other similar processes that have been studied.

We now present our results on explicit representation.

Given arbitrary values (aO,n, aj,,, ..., a.,.) and (b2,., b3,., .., bn_1,n), we wish to find the explicit form of the polynomial R.(x) of degree < 2n - 2 such that (2.1.4) (2.1.6) hold. For R.(x) we evidently have the form

n n-I
(2.1.7) R. (x) = E ak,f rk,l (x) + Y bk. Pk,. (X)
k=0 k=2

where the polynomials rk,.(x) and Pk,n(x) are the fundamental polynomials of the first and second kind. These polynomials are of degree < 2n - 2 and are uniquely determined by the following conditions.

(2.1.8) -rk, n(tj,= 0 for z k
k"n" , ~1 for =k'

k = 1, 2, ..., n- 1,

r o . n -=j ti. , n. ) : o (2.1.9) , (1) = (1) = 0
r0.n(i) = r=. (-1) = 0,

r., (xi,,) = rk., (ï¿½ 1) = 0,

o,n

(-.) = r',(1) = , (Xi n =rf,.n (Xi,.n) = 0

o for k
(2.1.10) Pk, (tj,.) = Pk,n(+) = 0, P,(xi,) = for i k"
k = 2, 3, . , n - 1,
(j = 1, 2, ..., n - 1 and i = 2, 3, ...,n-i).

The following theorem presents the explicit representation of these fundamental polynomials.

Theorem 2.1

The fundamental polynomials Pk,n(x) and rk,f(x) are given by

Pk., (X) = -P( (x) -Xk
n (n -I) P,- 1 (xk,)

(2.1.11)

n-1 (2r-1) n, (xkfl) r (x)
r-2 r(r-l) [n(n-1) + r(r-)] k = 2, 3, ..., n - 1, (2.1.12) rkfl(x) = Ak.n(x) - Bkn(x) , k = 1, 2, ..., n - 1, where

and

Ak n (x) = Bkn (x) =

-1 (2 r - 1) P r -1 ( t k,) 1C, (X ) r-2 n(n-1) + r(r-1)

r W (x) = r.n (-x) = (1 +x) P-1(X) P,-(X)
2

(2.1.13)

+P( n-2 (2r-1) 7c,(x)
+ z --I (X) E-1
r-2 n(n-l) + r(r-l)

+ (n-2) n-1(x) 79a-1(x)
2(n-i)2

The relative simplicity of these explicit forms is crucial to proving the convergence results in the next chapter.

(1 -X2) P n-1 iX) pn' X

2n (n - 1 ) Pn-, (x)
(1i- t2,, 2 ]3

Preliminaries
Here, we list various known results used in the proofs of the next section. The following identities were taken from Chapter Three of Sansone [32]. (2.2.1) (1-x2) P1-1(x) - 2x P,'_1(x) +n(n-l) P_1(x) = 0 (2.2.2) (1-x2) PJI//_I (x) - 4xP/,/_1 (x) + (n-2) (n+1) P'-- (x) = 0

(2.2.3) 71(x) = -r(r-l) P1 W(x) (2.2.4) (1-x2) P1. (x) = (r-l) P1-2 (x) - (r-1)XPr_1(x) (2.2.5) xP 1 (x) - P - 2 (x) = (r-l) P,-1 (X) (2.2.6) Pr-i(X) - XP/_2 (x) = (r-) P1-2 (x) (2.2.7) Pr_1i(x) - PI-3 (x) = (2r-3) P-2(X)

(2.2.8) (r-2) PI-3 (X) - PI-2 (X) ( P( l-x .1P-2 (X) + P -3 (X) MZ (2s-1) P,I (x) P._1 (y)
S=1
(2.2.9)
P'r (x) P, (y) - P1 (x) P'_1 (y)
y-X

The above identity is known as the Christoffel formula of summation. Differentiating (2.2.9) once gives the following.

MZ (2s-1) P,_-1 (x) P,,-, (y)
3=2

(2.2.10) = [P- (x) Pr (YV)y - P" (X) P'-j (Y) yV-x

+ P'1 (x) P, (Y) - P (x) P'_1 (y) (y-x)2 J

From Szeg6 [38], we write

(2.2.11) E' (2S-1) P.-1 (X) P,- (y) _ Pr (X) P-1 (y) - Prl (X) i4 (y)
s-2 s(s-1) x-y

Proof of Theorem 2.1
We first prove (2.1.11). In view of the uniqueness theorem, it is enough to verify that pk,.(x), as stated in (2.1.11), indeed satisfies the conditions in (2.1.10). The first condition clearly holds.

Fix any k E {2, 3, ..., n - 1}. From (2.2.1) and (2.2.3),

d2 I!(Xj~n
dJx2 [Pn-I (X) 71r (X) ]xx , 11 P-1 (Xj,n) 7Cr

(2.3.1) + P'{-1(xj.n) r, (xj,) = -r (r-l) P,-, (XJn) P1-1 (Xj;n)

- n (n-l) P_1 (xj,,) P1 (xj,n,) Hence,

Pk,, (xj,,)

' i-Xk~a ,I(ja
(2.3.2) n(n -1) P31(xk,

n-1 (2r-I) .P1'- (xk,))
T - 1 (x )xj,n)
Z-2 r (z -1)"

From (2.2.11), we now have

P.,n (x,.=)
(1 2 I P2X[.n) Pa-i (Xk, n) P-2 (Xk,n) Pn-1 (Xj. )
(2.3.3) n (n - ) 1 Pn3- (xk.=,)

(x .=(xkf)) 1
xj . - J~ ) P '- l (x ,.=)

From (2.2.1) and (2.2.5),

(2.3.4) P'(X P- (xj,)
k, P.-1 (Xk,f) (xj,n - Xkn) Pn1 (xk,

Recalling the fundamental functions of Lagrange interpolation, we see that (2.35) ( ) = 0 for j * k and

(2.3.6) Pk,n(xkn) = 1. We conclude, that all conditions in (2.1.10) hold.

Next, we turn our attention to (2.1.12). Again, we
need only verify the conditions in (2.1.8) hold, and all but the second derivative conditions are clear from (2.1.12).

Fix any k E {i, 2, ..., n - 1. From (2.1.12), we have (2.3.7) Ak. n(x) = I (x) ik,n (X) where (1-tk.) Pn1-1 (tk.,)

ikn (x) = Pf-1 (x)
(x- tkfn) P,-l (tkfl)

Next, we differentiate twice to get

-2n (n-1) P,,-, Cxj,,) ikn (xj,,,)
AkD(x3~n) = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(1 , ) PI-1 (tk,,)
(2.3.8)
2(n (n-- 1,2l)]-1 (xj,t)

Now, from (2.3.1), we have

(2.3.9) = -2n (n -1) Pn-1 (xj,.,)

n-I
-i (2-r-1) P,-1 (tk, n) /-l (xj,,).
r=2

We first apply (2.2.10) and secondly (2.2.4) to (2.3.9) and obtain

n (x3, 2n(n-1)2 P2-1 (xj.) P.-2 (tk,.)
2 2 [pl t, ]3 ( kn) (I -: [tk2-n j,~) x.-t (2.3.10)
= 2n (n-l) P2P-1 (xj,n)

(1 - t, [ t) ]2 (xj, - tk, ) Hence,

(2.3.11) 4 n . .) = A,(x .) - B1,'n(xj.) = 0 for j = 2, 3, ..., n- i. We conclude that all conditions in (2.1.8) hold.

Lastly, we prove (2.1.13). We begin by observing that r0..(x) = r.,.(-x) follows from uniqueness and symmetry of the zeros of P1 (x) and P -,(x) ï¿½ We, therefore, need only

verify conditions (2.1.9). Specifically, we show that r;,ll(jo) = 0 for j = 2, 3, ..., n - 1.

From (2.3.1), (2.2.10), and finally (2.2.8),
d2 n-2 (2z-l1) P,-, (X) 7c, (X)]

- E
41 -Pr-2 (n ) + (- 1) Pl (xl,).n 1 n-2
-P. ( M (2zr- 1 ) P _ (xj ,) P' -1 (1 )

(2.3.12) -(n-2) P1(xj,,) P1-3 (Xj,2) - P-2 (xj,.) [ 1 -xj.a

+ P1.-3 (x,.) - P.-2 (xg,.)] (1 -x 2) 2
P . -1 (X I )P -2 ,e - (d.
( xj12) [(n-3) P2 (xj1,1) - (n-l) P.3

We now apply (2.3.1) and (2.3.10) to (2.1.13) and obtain

2wn [pï¿½l-2 (x

+ P.- (xj,.) '!2 (Xj,)] + P._ (xP - ) P.-2 (Xj,.) + ... . ' [2 (x( -) - (n-i)P _ (x 1))]
1-x Pun .

- (n-2) P.-1 (xj,) P.2 (X,,)

(2.3.13)

Hence,

(1-x.) ", (x) P. I(xj .) rn .n .j '

(2.3.14)

P+ P=1 (xj,) P1n-2 (xj,.)
2P.-1 (xj,.)

+ (1-xj,,) P,-2 (X2,,) + (n-3) P-2 (x,) - (n-1) Pp_3 (xjn)

- (n-2) (1-xj,.) P.'-2 (xj,n) From (2.2.1),

(2.3.15)

(i-x~a) [P/I1 (xj,n) Pn-2 (xj,,) + Pn-1 (xj,n) Pn-2 (xJn)] = 2P1 (Xin) [x3,o Pn2 (X,n,) - (n-)2 P2x,)

We now apply (2.3.15) to (2.3.14) and obtain

( 1 - x , ) z , (,)
P . 1 (x j,j2) nr n,.

(2.3.16)

= (n-2) x,,. P.-2 (x3,,) - (n-1) 2 P, -2 (x.,)

- (n-1) Pn3 (x,.)

From (2.2.6) and (2.2.7),

(2.3.17)

(1 -x , ) " (j=
P n1 (x .) /, (X = 0

Hence,

(2.3.18)

r,, (xj,,) = 0 for j = 2, 3,

n - 1.

This concludes the proof of (2.1.13) and, consequently, the proof of Theorem 2.1.

CHAPTER THREE CONVERGENCE RESULTS FOR A (0;2) PROCESS

Introduction and Main Results
Let f be a real valued function defined on the interval [-1,1]. We now define the linear operator R,(f;x) by

n
(3.1.1) R"(f;x) = M f(tkfl) rkfl(x) k=O

where

ï¿½klX)- (1-x2) P~ol(x) P_,(x)
Zk.. (X) P" = (3.1.2) ( k,) ( -t., P - t,,
2n(n-l) P,_(x) n-1 (2r-l) P,-1 (tk)7c(x)
(1-tk.,) 2[pl_ n n(n-1) +r (x-1)

for k = 1, 2, ..., n - 1 and

r..n(x) = ron(-x) (l+X)
2
" n~ X)n- 2 - X 1)- x(X) (3.1.3) +EaI x X-2 n(n-1) + r(r-1) + (n-2)
2 (n-1)2 - (x) - (x)

The formulas (3.1.2) and (3.1.3) are taken from Theorem

2.1 and the tk.. are defined as in (2.1.1).

We now present the main objective of this chapter. Theorem 3.1

Let f be a continuous function on the interval [-1,1]. Then

(3.1.4) R,(f;x) - f(x) 1 ! C c (f; inX)

where C is a constant independent of f, x, and n, and w(f;6) represents the usual modulus of continuity.

We remark that Theorem 3.1 is a discrete interpolatory example of the Teljakovskii Theorem (Theorem 1.5).

Preliminaries
This section is comprised of a listing of known results which are necessary in the subsequent sections. All constants, stated or implied, are independent of x, k, and n. Also, we define x = cose and tk,n = cosek,n where tk,,'s are as defihed in (2.1.1).

We begin with the formula for summation by parts.

m -1
(3.2.1) Z aJb. = Z Ak (bk-bk.1) + AsbM where Ak = Z a,
k=M k=m r-m

Recalling the definition of the tk0.'s, we have [38]
(3 2 2) ( 2 ) k 2[ ]
ï¿½3.2.21 -tk.) > d, (n-1)2' k = 1, 2, ..., ,

(3.2.3) (l-tk, ) > d (n-1)2' k = [2] + , ..., n- ,

4d3 (n-i)2
(3.2.4) IP11-1 (tk,.) I d3 (n-12, k =

d4 (n-i)2
(3.2.5) IPI1(tk,n) I (n-k)3/2 k =

1, 2, ...,

[n 1 .. n -1

(3.2.6) Iek.n - Ok+l I > d5
n

Note (3.2.6) is a direct result of Bruns' Inequality. The following inequalities may be found in Szeg6 [38].

(3.2.7) IP, (x) I =

0( 1_ for -1 < x <1
kV'n sin 0)

(3.2.8) IP,(x)I =Q(i) for -i < x < 1 (3.2.9) IPn(x) ( 0) for -1 < x <

(3.2.10)

Ipi(x)n = ) for -1 < x < 1

From (3.2.2) - (3.2.5),

n3/2 ), k = 1, 2,

n - i.

(3.2.11)

2~ 1p1_ / tk )3

From (3.2.9) - (3.2.11),

(3.2.12)

o(sin3/2 Ok'j, k =
:On

The subsequent two results can be found in a paper by Szabodos and Varma [36].

IP (x) + P , (X) I= 0 ( sin@I Pn1iX) + Pn 1(X) =0 sIn

for -1 x < 1
2

for -1 < x < 1
2

From (2.2.8) and (3.2.8),

IP"(x) +P"+1(x)l =O (n) for-1 < x : 1.
2

From (2.2.4) and (3.2.9), we may write

IP.-2 (tk,,,) = 0( sin Okt ), k = 1, 2, ..., n- 1

Finally, from (3.2.2) and (3.2.3),

E 1 - 0(1) I e-et. . n2 sin2 le-ek,. I
n 2

(3.2.13) (3.2.14)

(3.2.15)

(3.2.16)

(3.2.17)

and

(1 - t 2 ! pI

(1 - t, ')2 13
k Pn-1 (tk.n)
(1 _ t2 .) 4 16l

1, 2, ..., n-1.

n3 sin3 16 (ko-l
2

From Erdos [12],

P (X) = 0 (1) (x- tkn) Pn-I (tkfl)

Lemmas

Lemma 3.1

11
For -1 < x : - and -1 < y g
22

we have

-P,(x) P- (Y)I = 0( +sin )

and

(3.3.2) P-i (X) Pr (Y) -

where x = cos 0 and y = cos 4.

(3.2.18)

Io-8k. I

(3.2.19)

(3.3.1 I [Pr - i x) P, (y)

P, X ,,(Y

Proof :

r jP_1, (x) P, (Y) - P, (x) PR-j (Y)

= r [P=- (x) P1 (y) + P, x) P, (y)] - [P1 (x) P, (y) + Pr (x) P._ (y)]

r 1P_- (x) + Pr (x) I I Pr (Y) I + r IP (y) + P_1 (y) I IPr(x) I

i-n Fi I T__

(sin 6 + sin
Vsn0 si-ng)-

In the above, we used estimates (3.2.7) and (3.2.13). To prove (3.3.2), we use estimates (3.2.8) and (3.2.13). jPr - (x) Pr (y) - P1 (x) P-1 (y) I : I P-1 (x) + Pr (x) I I P, (Y) I

+ I P, (y) + P, 1 (y) I I Px (x) I =

o( sinG + sin J

Lemma 3.2

1
For -1 < x < - and
2

-1 < y <

1

we have

0 ( sin + sin9)

and

(3.3.4) /P[.1(X)P1(y)

V(sin0 + sin )
( sin e/sinp J

where x = cos 0 and y = cos *.

(3.3.3) 1P."i(X) Pr (y) - Pr(X) P,_1 (y) I

Proof:

IPA)l W Pr(Y) - P"(x) P_1(l j[P"- (x) P, (Y) + P x) P, (Y)]

- [Pr' (x) Pr (y) + P (x) Pr1 (Y)] j 4P/ (x) + P"(x) I IP, (Y)

+ I P(Y) + PI-1 (Y) I Pr(x)! = 1

+ sin.1
s in3O)

O( sinO + sinVsin3O sin,

Here, we used the estimates (3.2.7), (3.2.9), (3.2.13), and (3.2.14). To prove (3.3.4), we use (3.2.7), (3.2.10), (3.2.13), and (3.2.15).

PI1 (X) P, (Y) - Pr(X) P-j (Y) 1 1 (X) + P" (X) IP, (Y)

+ I P. (y) + Pr(y) I () 4 (X)

= sn + ~sin)

=O(V- (sine + sinO)

Lemma 3.3

' [ 1- (X) P 5 (tlf) - PS (X) P._1 (tk,,)]
s=2

r ( P -x ) P(j t , ) ,+ 1

- P (X) P, r Pr-l (X) Pr - Pr (X) P1- (tk,n) tk,n - X

Proof:

S[z'I-1 (X) P5 (tk,,) - P~ x , 1 f)

- I- (X) Ps (tkfl) - E P8-2 (X) P,-1 (tk,,)
S=2 s=2
r

- (2S - 1) P5... (X) PsI (tk,fl)
S-2

r -1
PI-1 (X) PS(tkfl) - PI. (X) P- tk)
a-2 s=2

r (2S - 1) P-I (X) Ps-1 (tkfl)
s=2
P (x) P (t) - (X) P, (tkf,n) - P (X) P (tkfl)
,r-1 (X) L (tk,n - X

The above equalities used (2.2.7) and (2.2.9). We next apply (2.2.8) to obtain

E [P/1 (x) P, (tkf) - P" (x) P,1 (tk,) ]
s-2

-P (x ) + r 1 X I P (xtk))

[Z PI-I (X) Pr (tkn,) -, P(X) P,1 (tk,)1 k,n - X

The lemma follows+.

Lemma 3.4

E 2s2 [P,-I (x) P, (tk.,) - P, (x) P,_- (tk.)]
a"2

(i - x2) (r + 1) p/ (X) Pr 1 (tka) - P. (X) P1 (tkn) tkn - X

+ Pr (X) P+l (tk,n) - Prl (X) Pr (tkfn) 1 (3.3.5) (tk

(tk,) P,1 (tk.,) P, (x) (1 tkl PI t.)P x
k, X - tk.zz

+ P1 (tk..) P1, (x) - Pr1 (tk.) P, (X) 1 (X -tk) 2j

- (r + 1) r[P, (x) P,. (tk,,) - P,., (x) P. (tk,fn)] + 2 (x - tkn)

Proof: From (2.2.4), we may write

r(2s + 1) s [P,-, (x) P8 (tkl) - P5 (x) P,-l (tk.n)]
s=2

( 2s + 1) P (tk.o) [ (1 - X2) P. (X) + sxP8 (x)]
8=2

)(2s + 1) P )(/ ( tkl) + S tk, P. (tkl)1
s-2
(3.3.6)
I
2( -X-) 1: (2s + 1) Ps(tkfl) P/,(x) 8=2

-)tk ( (2s + 1) P. (x) P, (tkl) s-2
r
+(X- tk.,n) E (2s + 1) s Ps(x) P(tkl) s-2

42

We now work with the last sum in (3.3.6). From summation by parts (3.2.1),

Z (2S + 1) sP.(x)P,(tk.l)
v-2

[S- (S+1)] (2M + 1) P.(X)Pm(tkfl)
s-2 z-2

+ r E (2m + 1) P. (x) P. (tk_.)
m-2

(3.3.7) = - 1 (sP ) (X) P.+ (tk.) -Pa+i (X) P (tk,fl)

X-1
+ 2 (1 + 3xtk,,) + r(r +)
s=2

Pr (x) Prl (ktkn- Pr+ (X) Pr (tkn,'

- (l1 + 3xtkf,) .

We used (2.2.9) in the second equality of (3.3.7). Now,

r
(x - tkn) Z- (2s + 1) s P, (x) P, ( tk,n) s-2
r

S2S [P-1 (X) P.( tk, ) P. (X) P- (tk)
8-2

-2 [Pi (X) P2 ( k, n) - P2 (x) Pi ( tk,.)]

+ (x - tk,,) (r - 2) (1 + 3xtkn)

(3.3.8) - r(r + 1) [P1 (x) Pï¿½+(tk.) - (X) Pr(tk.)]

- (x - tkf) r (1 + 3xtkfl)

S I P-(X) P(tkl) - P, (X) Ps-1 ( tkn)]
s=2

- (x- tkf) ( = 3xtk.l)

- r (r + 1) [P, (x) P1+1 (tkf.) - P,+1 (x) P, (tk.,)].

Together (3.3.6) and (3.3.8) imply that

Z (2s + 1) s [P-1 (x) P, (tk,,) - P8 (X) P-i (tkn)]
s-2

S [P-1 (X) P8 (tk) Ps (; Ps-) (tkfn)]
s=2

(3.3.9)

z
(1 -x2) E (2s + 1) PS(tk,,) P"(x)
s=2

(II
tk-() Z (2S + 1) P (x) P' (tk,.f) s-2

- (X - tkn) (1 + 3Xtk,n)

- r(r + 1) [Pr (x) Pr+1 (tkfl) - P1+1 (x) P, (tk) ]

With further calculations on (3.3.9), we obtain

r
E 2S2 [Ps1 (x) P, (tkn) - P8 (X) P.-1 (tkl)]
s=2

r+1
( - X2) (2s - 1) P_1 (tkfl) P/ (X) 8=2

(3.3.10)

2 r+1 tk, ) - (2S - ) P,-1 (x) P,1-1 (tkl)
s-2

-3(i -X2) kn + 3 (1 - tk, )X

- (X - tk.n) (1 + 3xtk,,)

- r (r + 1) [PI (x)p , (tk,) - P1 1 (x) P, (tkf) ].

From (2.2.10),

Z 2s2 [P,-. (x) P, (tk,.) - p, (x) p,_1 (tk.l)]
s-2

S(-x2) (r + 1) (x) P 1-1 (tk,.) - P2-i (X) P, (tkfl) tk,n - X

+ P1 (x) P.1 (tk,.) - P+l (x) P, (tk,.) (3.3.11) (tkfl X)2 S(tk) P1-1 (X) PI (tk,,) Pr (x)

+ P-(tk..) P,-1 (X) - Pr11 (tk,,) P, (x) j+2 (x (x - tk- .) 2
-r(r + 1) [Pr (x) Pr,1 (tk.) - P, (x) P1 (tk,a) ]

The proof of the lemma is finished.

Lemma 3.5
For -1 < x < 0,

1tka2 E 21 _ X 2s2 [P,.1(x) P3 (tkD)
l k.-xI + A l --2

- P,(x) P,_1 (tk.)]I

(3.3.12) = 0( 1 1
[n~sinUsinok . n~sin2 le -Fk,--I ï¿½ 2

+ n3sin+ie-k._ ] for i = 3, 4.
2

Proof: From Lemma 3.4,

M 2s2 [P._ (x) P, (tkf) - P5 (x) P., (tkn) ]

tk, (1 x2) 1 IP/ (X) P,-1 (tk,f) - PI+1 Wx P, (tk.f) I
itk2 n -'

+ (1 X2)(Z + 1)
trk,. - X12 lp. (X) P*1 ( tk,.) - Pr-1 (X) P, ( tk,.)I

(3.3.13)

+ (1 - . (rl) ,(tk ) P1 (x)
I t2 - -xl

-P '+ (tn) P, (x)

(1 -tk, .) (Z + I
+ (2 . - X12 IP (tk) P,'1 (x) - Pr+l (tk,,) P, (x) I

+ r Cr + 1)I1P, Wx)P., ( tk..) - P,,, Wx)P,( tk,=) I + 2 1 x- tk,,21

We now break into two cases. Case 1: -1 < tk,n 1
2

To get (3.3.12), we need to analyze the order of the

following terms.

z=2 r 2n 4

From Lemma 3.2 (3.3.3),

__J (1 - x2) (r +)
n i ) I tk,, - x12

pZI/ (X) P1,1 (tkf) - P1+1 (x) P, (tk,,)

(3.3.14)

sin2O (r + 1) (sinO + sink,n)
I xk,-X2 - siP n3sinok,n

n(sin2 10-k.-I VIlff-sfink,.

and

=0.

tk'( + 1
12n 4 n 5 I1tk, -X12

IP- (t,,,) P,-, (X) - Pg.,. (tk,,) P, Wx

(3.3.15)

sin2)kn (r + 1) (sinO + sin~ko.)
I k, - x12 Sin ek,.sin8f

0( 1
= 0 3sin2 le- 0k.l VsinesinOk) 2 k.

From Lemma 3.1 (3.3.1),

(i (1 -x2) (+ 1)
W- ( 2 1n 1 n- k) I P1,.1 ( P - X (l3
* JPr (x) Pr-i (tk,.,) - Pr-, (X) Pr (tk.,n)

(3.3.16)

sin2O (r + 1) (sinO + sinko) I k,n - X13 r VsinSslnok,n

0( 1
= 0(s3 le- 6k,. VsinlsinJ

(15) tnrk. - X13

"pr (tk, ) P*1 (x) - P1,l (tk) P1 (X)

(3.3.17)

= 0 ( n 4 (-2 z n I tk,n - X13 r sin--snOUko.

0( I )
=0 n~sin 3 le - e,.l sit n),

and

xni

0 -L 1 +
n 4 2
r-2 ( r n

- _ r (r+l) z-2 ( 2n 4 ns ) tkn - x1

jP, (X) PV+1 (tk.l) - P+i (x) Pr (tk,)

= ( l ) r(zl+ 1) (sinO+ sinekcn) n- xI r ___S__0 ( (n 3sin 2 106 k,.1 Vy7s innk,

2 1 + (2) = 0 z= 22 n -4ns n 2

=0 (n 2 Vsinesinffk-

In case 1, the lemma follows.

Case 2: 1 tk, <
Since -- 2
Since -1 < x <,, 0, we note that I tk,. - X I Z! An elementary application of estimates (3.2.8) and (3.2.10) along with some calculations on the various terms in (3.3.14) - (3.3.19) gives

(3.3.18)

Also,

(3.3.19)

1 i 1
1 tk,nX1ZXI ( r2n4

(3.3.20)

M 2s [PS_1(X) P,(tk, n) - P,(x) Ps_1(tkn)

2 1 n2 Vg/HSnk,nl

The lemma follows. Lemma 3.6

For -1 < x < 0,

1
n41tkf. - xI

n-j
2s2[P1 (x) Ps (tkfn) - Ps (X) P,-1 ( tk, n)]

(3.3.21)

nsinosink,n [2sin2
,n n si

+ 1
n3 sin3 n - + . lI
2

) for i = 2, 3.

0 n2(1)

49
Proof: We begin by utilizing Lemma 3.4 as in (3.3.13).

7- 2S2 [P._1 (X) P. (tk ) - P, (x) P,.1 (tk,.)
5=2
< I-X2) (n -i+ I) ip_(x) P- t I tk.- XI

-P1+1 (X) P.i (tkf) (1 - x2) (n - i + 1) I t,. - x12
ï¿½ P.-i (X) P.-i-1 (tk..) - P.-i.1 (X) P.-i (tk..)

(3.3.22) (1 (n-i+1)
I tk.- XT I

(I (1 -,) (n - i +
- i+1 (tk,,) P,-1 (X) I (1- I - X12 * P.-I (tk".) P.1i- (x) -ill1 (tk..) P..1 (X)
+ (n - i) (n - i + 1)P-i (X). P-i+1 (tk,)

- P.-1. (X) P.-1 (tkfl) + 2 1fx - tkn I We break into two cases. Case 1: l1 2

As in the previous lemma, we need to estimate six different terms. From Lemma 3.2 (3.3.3),

( - X2) (n - + 1) )
- l Itkn X I +
(3 3. 3 IP.i ( ) Pn. x .-. =.)P-i (X) I

0

Ai e (n3sin 8 emml wineed oetmes and

(1i-tk,) (n - i + 1) itkf)Pf-iz x
n' I tk" - X12 1Ii(k, ,il(X

(3.3.24)

- P-i1+ (tkn) P.-i (X)I

= 1 ( S 8 , 1s i nl s i nl k n n3 sin2 le-elVlasl

From Lemma 3.1 (3.3.1),

( - x2) (n - i + 1)
n4 1tk, n _ x13 jPn-i (x) P-il+ (tk,n)

(3.3.25)

- pli+1 (x) P-i (tkfl) I

0( 1
= 0(4sin3 le-ek, ,I sinosinU , )

(1- t, (n - i + 1)
n4j tk,, - x13 [Pn-i (tkn) P,-i-1 (X)

(3.3.26)

- Pn-il (tkn) Pn-i (X)I

= 0 (- ekosnl 8sinsin ,
n2 si) I

and

(n - i) (n - i + 1) 1Pa-i (x) P-i+1 (tkn)
n1l tkn - XI

(3.3.27)

- P-i1 (x) P,_i (tk,) I

0 n - eknI IsinSinlk,n
n3sin 2 snjinf,

Also,

(3.3.28)

2 C
n4 ~ V n1 sn i~~

The lemma in case 1 follows.

Case 2: 1g tJ, <1
2

Here we note that

I tk'n - XI >

-1 and apply (3.2.8) and

(3.2.10) to the various terms in (3.3.23) - (3.3.28). The result is

1
n4 Itk., - XI

n- 2s2[P8_1 (x) P8 ( tkfl) - P. (x) P-1 (tk.) ]
8=2 LII

(3.3.29)

(n2 Vsinsinkl)

and the proof of the lemma is complete.

Lemma 3.7

For -1 < x < 0,

X- tk, I Irk,. (x) i = o ( sinO

(3.3.30)

+ 1 n3 sin3 Ie - ek,.
2

+ i]) for k = 1, 2,..., n-i.
n

Proof: To begin with, define

at,= = n(n - 1) + r(r - 1) and Lz,.

1

From summation by parts (3.2.1) and (2.2.10),

n-I (2r - 1) Pr_1 (tk,,) Pz-i (x)
z-2 n(n -) + (r-i)

E[1,r,n- 1.] E (2s - 1) P-I (tkfn) P (
r=2 s=2

+ -1.,1 E (2s - 1)P. 1 (t.) PA1 (x)
S-2

1-2 r2
Z-2 ï¿½ ï¿½ r 1 ,n r.l,n

i(X) P tkn) - PX -1 ( ]tkn)
tk,n - x

-2 2 Pr )1 (X)X) I- ( Pk , D
+2 r
X=2 a r . (tk1 - X) 2

- (n - 1) P'_1 (x) P,_2 ( +k.f) P.-I (x) P.-2 (t,.) 1
2(n - i)2 tk,. - x (t .n - x)2 j

Recalling the definition of rkn(X) given in (2.1.12), we may write

(x - tk,.) rk, (x) = Z1 + Z2 + Z3 + Z4 + Z5

(3.3.31)

(3.3.32)

(3.3.33)

where

1I n2 sin2 le - ek-.l
2

4n(n - 1) (1 - x2) P"-1 (x)
(1 - t )2pi ( tkn) ] 3

(3.3.34)

n-2 12
: ~ ~ ~ [:'-1 (X x, _t , I).o - pr ._., , ]
X=2 l r,nO~xl,n

-4n (n - 1) (1 - x2) Pn-1 (x)
(1 tk,,)2 (p"_- ( tk. ) ] 3

r2_ Pr-1 (x) P (tk) - P (x) Pr-li (tk.l) ar,nctr+l,n[ X tk,n

(1 x2) P",1(x) PI_ (x)
t (1) [P-1 (tkn)]2

-n(1 - x2) p"_1 (x) (1 - tk2, n)2 [P"-l ( tkn)

.3 [P l-: (X) , (tkf)],

n (1 - x2) p_1 (x) (1 - tk. ) 2[(P"_l ( tk. n

[P. (X) P.-2 (tkn)
X - t, ,

We begin by working with Z2. From (3.2.1), estimates (3.2.7) and (3.2.11), and Lemmas 3.5 and 3.6,

Z, =

z2 =

(3.3.35)

n-2 r=2

(3.3.36)

(3.3.37)

Z4

and

(3.3.38)

z5 =

sin /2,= - 1-,1 1
I tk,,- X1 -2 gx n(Lr~~n Ccr*1, nax2,n

ï¿½ I 22[p._, (x) ,. (tk.) - P. (x) P ,- (tkfï¿½) +

1 n-2

+ nE 2s2[p (X) P, (tkl) P P, (x) P-, (tk.n)
&-22

8-+ 2S2[p_ [P. (x) (tk~) P. (X) P,~. ( tk)

1
n-2,2 an-1, n

0 sine 1 218k 1 n 2 sin2 ' -e l n3 sin3 le - ekl '1 S 2l' 2

We now work with Z,. From (3.2.1) and Lemma 3.3,

4n(n - 1) (1 - x2) Pn1 (x)
. (1 - tk.,) [_1 (t,,,) ]

r2-3 r-2 Lar~n (Zr+l n

(.r + 1)2,
lI+1.n cr+2,n

(3.3.40)

- E [P"_l (x) P, (tkf) - Pï¿½ (x) P,_ (t .) ] ï¿½ s-a2

+ (n - 2 )2 n= l + Q2+ 3+ Q4+ 5+ Q

- P' (x) P,- ( tkf,)] }

(3.3.39)

where

-4n (n - 1) (1 - x2) Pnl (x)
(1 _- 2 )2["_ t "

n-3 (2r + 1) [n(n-i) - r(r + I) x=2 a1r,na z+l,nr+2,n

. [ r(P((- (x) Pr ( n)

4n (n- 1) (1 - x2) P_12 (x)
(1 - tn)2[p1(tk, n) ] 3

+1]

n-31) r(+
3 (2r + 1) nn ) r[ z-2 ar,n (r+l,naz+r2,n

4n (n - 1) (1 - x2) Pn-1 (x)
(1 tn)[P-1 (tkn)]3

n-3
- ~ (2r + 1) r [(n - 1) - r(r + 1)] r=2 ar,nar+l,nx+2,n (tk,n - X)

" [P-1 (X) P, (tk,) - Pr (x) P-1. (tk,n)],

4n (2 - 1) (1 - X2) P"-1 (X) (n - 2)2

(1 - tk,n) [P-1 ( tkn)]3 an_2,n1 ,n

(n - 2) (P.-3 (x) - P1-2 (x))
1 - x P.-2 (tk,n)

Q1 =

(3.3.41)

(3.3.42)

(3.3.43)

Q4=

(3.3.44)

+ 1]~

PI (X) P, (tkn)

(3.3.45)

-4n (n - 1) (1 - x2) Pn1 (x) (n - 2)2
5 t.)[Pn-1 (tkn)]3 -n_2,n an-l,n

[Pn_2 (X) Po_2 tk,)]

-4n (n - 1) (1 -x2) Pn_ (x) (n-2) 3
06 (1 - t. ,n 2P~I- ( tkn) ] afl-2,fanl1,f .46)
SPn-3 (X) Pn-2 (tkn) - Pn-2 (X) Pn-3 ( tk, n)
[ tk,n - X

We now estimate Q1. From (3.2.7) and (3.2.11),

loll = 0 (sin'I2e Vsinlkn .47) n-3

X-2 n3 sinUsinkn n2

and

(3.3.

(3.3.

(3.3.48)

1- 2(2.r + 1) r [n (n -)-r(.r + 1)]
z-2 ax,nar+l,n a+2,n

[P-i (x) P, (tkn) - Pr (x) P,-1 (tkfl) ]

n-4 (2Z + 1) [n(n - 1) - r(r + 1)1
-2[ r ar, n ar+,nar+2,n

(2r + 3) [n(n - 1) - (r + 1) (r + 2)]1
(r + 1) ar+l,n(r+2,n(a+3,n

T I 2S2 [P,-i (x) P, (tk,l) - P, (X) P,-1 (tk,n)]
s-2

+ (2n - 5) [n(n - 1) - (n - 3) (n - 2)]
(n - 3) an_3,ann2,nn-,n

n-3
Z 2s2[P8_1 (x) P8 (tk.n) - P8 (x) P-1 ( tk.) ].
s=2

Together, (3.3.43) and (3.3.48) imply that

(3.3.49)

n2(1 -x2) IP'_1(x)1
2 1 0 (1 - tkn)21p.1_(1tk ) 13

+_ + ) 1:2 2s2 [P, . 1 x) P, (tk) - P (X) P,_ (tkf. )

+ 22P.lxP, (tk, n) - P X I(t

From (3.2.7), (3.2.11), Lemmas 3.5 and 3.6, along with (3.3.49),

(3.3.50)

I Q31 = 0 (s ine

+
n3 sin3 16 - ek,
2

1I n 2sin 2 10 - Ok,.l
2

,+ f

Using (2.2.4), we may rewrite Z3 as

(3.3.51)

Z3_

(n - 2) (1 - x2) Pn-2 (tkn) Pn-1 (x) Pn-1 (x)

We now look at the order of the following combination of terms.

Z3 + Z+ = (1 - x2) P.-2 (tk.) P.-i (x)
(1-tk2,) 2 (p.'_, ( tk. ] 3

(n i) P'-1 (x) - nP,- (x) = -(1 - x2) P.-2 (t,.) P.-1 (x)
(I tk2,,,) 2[ p.'_ (tk,")]3

4n(n - 1) (n - 2) P-2 (x)
Cn-2.n n-1,n

r -n )P2-2 (x) 1
-' (x) + P-2 (x)) + (-2n + 3)
(n - 1) (n2 - 3n + 3)

(3.3.52)

(1 - tkn) [pn'- ( tkn 3

From (3.2.7), (3.2.14), and (3.2.11),
(1 - x2) I PI-2 ( tkk,) P,1 (x) I
i -tk .) 2 1 p" l ( tk, n 13
(3.3.53)
i / / sinO). IPn-1 (x) + Pn-2 x = 2 (-iFrom (3.2.8), (3.2.10), and (3.2.11)

(1 - X2) I P.-2 ( tk,.) P1-1 (x) I(2n - 3) P/.2 (X)
(1 -o.) 2I., (tk~n) 13 (n -1) (n2 - 3n + 3)

Hence,

jz3 + z4 + 95 =0-(sine).
n2-

We next look at another combination of terms.

n (1 - x2) P'_1 (x)

["+-i
Z5 + Q

(X tk.,)( tk, [ Pn-1 (tk, n)]

(X) P-2 ( tk, n) + 4(n -1) (n -2) an-2,n Cn-I,n

* (P.-3 (x) P.-2 (tk,.) - PI-2 (x) Pn-3 (tk,.l))

n(1 - x2) P'_1 (x)
( x )- tkP ) (1 tk + 2 p "_ ( tk fl3 [P"-_. (x) Pn_2 irk,.) + Pn-3 (X) .2(k)

- Pn-2 (x) Pn_3 (tkl) +

(-2n2 + 6n - 5)
(n - 1) (n2 -3n + 3)

(P._3 (x ) P -2 (tk, ) - P -2 (x ) P -3 (tk l) ) .]

(3.3.54)

(3.3.55)

(3.3.56)

= ( sin8)

60
From (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1), we have

n(1 - x2) IPl (x) I 1-2n 2 + 6n -51
1jX - tkn1 (1 k2 t) 2 1p_ (tkf.) 13 (n - 1) (n 2 - 3n +3)

1 P,_3 (x) Pn-2 (tkf) - P.-2 (x) PI-3 ( tk,l) I

sin2e/sinjk,n (sine + sinkk,)
Ix - tk,n I n 3 sin Onkn )

sine I 01 -1 < tkn
n3)sin le-ek.l
2

1
2

From (3.2.7), (3.2.8), and (3.2.11),

n(1 - x2) I P, . (x) I I-2n2 + 6n -51

X- tk,1 (1 - tk, 2 )1 p-. (tk) 13 (n - 1) (n2 - 3n +3)

(3.3.58)

.I P,-3 (x) P._2 ( tk,) - P,-2 (x) P,-3 ( tkf,) I

= 0 ( sin3/2eosin kn)n

n(sio)

for I k,
2o -1:! tk~.n < 1.

From the following identity [32],

(3.3.59)

(n - 2)P11_3 (X) = -(n - 1) P,-1 (x) + (2n - 3) xPn_2 (x) ,

we may write

P, 3 (x) P.-2 (t .) - Pn-2 (X) Pn-3 ( tk,)

- (2n -3) (X
(n - 2)

(n - 2) P,-2 (tk,.) Pn-1 (x) (n 2)

(3.3.60)

- tkf) P-2 ( tk,) PI-2 (x)

(3.3.57)

61
Hence,

n(I - x-) P,-1 (x)
2 )2 3
(x - tkn) (1 - 2 ['pk1 (tk0f) ]

[Pn-l (x) Pn-2 (tkn) + Pn-3 (x) Pn-2 (tk.l)

Pz-2 (x) Pn-3 (tk.l) ] ( 3 . 3 . 6 1 )n i x P - ( )

(x - tk,) (1 - 2 [pl_ (tkl) 3

-12 P 1(x) Pn-2 (tk,n) + (2n --3)
L(nl2) (n -2)
* (x - k,) P-2 (tk,) P-2 (x) ]

From (3.2.7), (3.2.11), and (3.2.16),

n(l x2) P -1 (X) I Pn2 ( tkl) I
tk ( - tkn)2IP1 (tk,) (n -2)

(3.3.62) 0=o SOin2OSine k'1
n3sin IX - tk,n
= sine
n3 sin lo - Oko.l)
2

From (3.2.7) and (3.2.11),

n(1 - x2) IP1-1 (x) P-2 (tkfl) P,-2 (x) I (2n - 3)
(1 _ t: 2 ) 2 1 p"_ ( :" )1
( , P_(tk,) (n -2) (3.3.63)

W0 (sine)
n 2

We conclude that

(3.3.64)

1Z5 + C)6 0 o(sinO[2i1kn
n n2 sins 113 - ekï¿½,.
2

To estimate Q4, we use (3.2.7) and (3.2.11) to obtain

(3.3.65)

We now observe that we need only estimate Q2. From

summation by parts (3.2.1) and (2.2.10),

IQ41= o sinO)

(3.3.66)

y3 (2r + 1) [n(n 1) - r(r + 1)] p'(x)P(tf) r - 2 / rr (rlï¿½n,

n-: [ n(n- I rr + - I) - (r + 1) (r + 2)1
r-2 a,,nar~l,n(Xr 2,n ar~l,n r+2,n r+3on

(2s + 1)P(X)P,(tk) + n(n-i) - (n - 3) (n -2) a-2 an_3,ncn_2,nan-l,n

E (2s + 1) PW(x)P(tt,)
s-2
n-" 4 (r + 1) [2n(n - 1) - r(r + 2)]
x-2 Mrn Zr 2-,n ar+2,n Cgz3,n

* (z + 1) P' (x) P +, (tk.l) - P.l (X) P, (tkl) I tk,n - X

+ (r + ) Pr (X) Prl (tk'n) - Prl (X) Pr (tkn) - 3 tk (tk, - X) 2

+ ï¿½2 (2n - 3)
an-3,a n-2,n On-l,n

ï¿½ (n - 2) P2-3 (x) Pn2 (tkl) - P.2 (X) Pn-3 (tkfl) tk,. - X

+ (n -2) Pn-3 (X) P.-2 (tk, n) -_Pn-2 (X) Pn-3 (tk n) - 3tk. n.
(tkfl - X)2J

Now,

(3.3.67) Q2 = R, + R2 + R3 + R4 + R5 + R6 where

(3.3.68)

16n (n - 1) (1 - x2) P-1 (x)
(1 - t2,)2 P'(tkn)]3

r + [2n(n - ) - r(r + 2)] r=2 (Zr,n r+l, n ar+2, n ar+3, n

Pr (X) Pr + (tko,) -P1+ (X) P, (tl)
tk n - X

16n (n - 1) (1 - x2) Pn-1 (x)
R (1 - tk, ) [p"_, (tk ) ] I

n-4 (r + 1) 2[2n(n - 1) - r(r + 2)] (3.3.69) E=
=2 (X,n (r~l,n ar 2,nar 3, n

r(x) Pr-l (tk,n) - P (X) Pr (tk) 1 (tk - 2

-48n (n - 1) (1 - x2) P,-1 (X)
R3 = " l k
(1 - tk,n)[P-l(tkn)]3
(3.3.70)
n-4 (r + 1) [2n(n - 1) - r(r + 2)]
E tk,n, X=2 ar,n ar+l,n xr+2,n ar+3,n

8n (n - 1) (n - 2) (2n - 3) (1 - x2) Pn1 (x)
an_3,nan,nanin(I - t )2[Pn-1 (tk.n)]3 (3.3.71)
SP-3 (x) Pn-2 (tkf) - P-2 (X) Pn-3 (tk.l)
tkn - X

8n (n - 1) (n - 2) (2n - 3) (1 - x2) Pn-1 (x)
R5 =
an-3,nn_2,nOn-l,n (1 - tk,n
(3.3.72)
Pn-3 (X)P-tk.l) - P-2 (X) Pn-3 (tk.n)
(x - ( tk, - x) 2

and

(33.73) R -24n (n - i) (2n -3) (1 - x 2) Pn-1 (x) tk.n
n_3,nan_2,nnl,n (1 - 2 2 [/kl (tkfl) ] 3

We proceed to estimate R, - R6.

(3.2.11),

and Lemma 3.1 (3.3.2),

I12 = 0 (Insanekl sinO2

n-4
r=2

(3.3.74)

2 (Vsin~k, n + VrS-in0)
n6 rr tk.n - x12 )

sin2O (sin6 + sink)
n3l tkn - x12

0 ( sine f or -1
= n3sin2 le k,.I 2)

From (3.2.8) and (3.2.11),

(3.3.75)

I l ( (sine) for
nR1=0 2 2

kn1* tk, nK : "

tkn < 1.

We move to R3. From (3.2.7) and (3.2.11),

(3.3.76)

I1 --0 ( )in )

We next apply (3.2.8), (3.2.11), and Lemma 3.2 (3.3.4)

to R4.

From (3.2.8),

sinO (sinO + sink,n)
n3l tkI - xI

= ( I sin ) for -1 <
n3 sin I 8k( f. o
2

Also, from (3.2.8) and (3.2.11),

1R41 = O sin for 4 ! tk < 1

From (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2),

Rs[= o sin2O/sinOk(Vsinkol + vl--n) n4 ( tk, - x) 2

(3.3.79)

sin2O (sinO + sinOk.n)
n4l t X12

sinO for -i<2t.
n3sin2 le-e fok,-l 2
2

When-! 4 n < 1 we use (3.2.28) and (3.2.11) to get
Whn2

sinO)'
n 2

(3.3.80)

JR41 =-0

(3.3.77)

tk,n < 1.
2

(3.3.78)

JR51 = O (

From (3.2.8) and (3.2.11), (3.3.81) 1R61 =O(sine

Finally, we concentrate our efforts on R1. Define

(3.3.82)

- -2n2(n-1)2(2r+3) + 4n(n-1) (r+2) (3r2+7r+3) - 2r(r+l) (r+2)2(r+2)
Yr' 4~n "1n Cr*,na r+,a*4

and

a n (n-3)2 (n2+4n-8)
an-4,n an-3,n Cn-2,n an-1,n

From summation by parts (3.2.1) and Lemma 3.3,

(3.3.83)

24 (r + 1)2 [2n(n - 1) - r(z + 2)] .=2 (%z,n az+ln (r+2,n Cr+3,n

[P/r(X) Pr l (tk,l) - Pl.1 (X) P,(tkl) ]

1 - + 1)2 [2n(n - 1) - r(r + 2)
z-2[ az,naz+l,n tr+2,nr+3,n

_ r+ 2 )2 [2n (n - 1) - (Z + 1) (Z + 3)]
z+l,n gz+2,n ar+3,n Cgr+4,n

r
M [P I (x) P~ ( tk,fl) - P,'+i (x) P, ( tk,fl) S-2

+ (n - 3)2 [2n(n - 1) - (n - 4) (n - 2)]
n-4, n n-3,n n n-2,n an-1,n

n-4 /I
n-4 [P8 (x) P+1 (tk,l) - P+li (x) P. (tk.n) ]
s=2

n-5 r (P,_1(x) - P,(x)
r=2 11 - x

*+ 1 - /XPr Pr(tk,n)

P11 (X) P, (tk.,) - P, (X) Pr-1 (tkl)
-r 1 tk, n - X

+ n4, n (n-4) (P-_ (x) - Pn-4 (x))

Pn 4 (tkfl) + I - Pn-4 (X) Pn-4 (t:kn) - (n - 4)

Pn-S (X) P-4 ( tk.n) - Pn-4 (x) Pn-5 (tk-, ]
tk,n - X I.

69
We may now write
(3.3.84) R,= S + S2 + S3 + S4 + S5 + S6

where

=, 16 n(n - 1) (1 - x2) P1-1 (x)
(1 - t) 2[pIl (.tk,) ]3 (.tk - X)
(3.3.85)
n- [rr(P.-l (x) - PZ(x)
Z ,n 1. -xPz ( tk, + 1
r-2 '1-X

-16n(n - 1) (1 - x2) Pn1 (x)
(3.3.86) (1 - tkL) 2[p,11 (tkf) ]3 (tk,fl - X)
n-5
E yr nPlx) Pr (tkl) ï¿½
=2

16n(n - 1) (1 - x2) Pn-1 (x)
(3.3.87) (1 - t [PI (tkf) n - X)

n-5
E ZYr, n.rPz-1 (x) Pr (tk,n) - Pr (X) Pr-1 (tk,.) ] ,
Z-2

= 16n(n - 1) (1 - x2) Pn1 (x) (yn4,n
(1 - t [) 2[p_ (>tkf) ]3 (to - x) (3.3.88)
.rn -4) (P-5 (x) - P-4 (x)) ( (tk, n)]+ 1 1 -x

= -16n(n - 1) (1 - x2) Pn12 (X) an4,n (3.3.89) (1 tk.D)[Pn1(tkn) (n X)

ï¿½[P.-_4 (X) P_4 ( t,.o I)

and

-16n(n - 1) (1 - x2) Pn-1 (x) an_4,n

-t

(3.3.90)

(1 - t,') 2[p1_ (tk)]3 (tkf _ X) 2

[Pn-5 (X) PI-4 (tkn) - Pn-4 (X) Pn-S ( tkf)

We now estimate the order of the terms S

begin, notice that I :Y,,n 0 n( 5

From (3.2.7),

Is,
(3.3.91)

- S6. To
o( n4

and IC-.4, nI =

(3.2.8), and (3.2.11),

sin3/2Osinok, 1=I tk n - xT

2-5 r=2i

0 (_sine n 3 n sin le - k'.l "
2

Using (3.2.7),

(3.2.8), (3.2.10), and (3.2.11), one may

obtain

(3.3.92)

I s2l =o( sin2O sinkUn n-5
k,.- x( nï¿½ nsinNsink.

o sineO
n 3 n sin 2 le - (k.-I
2

We apply (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2), to

derive

S =

71
IS31= OI n sin'e~sinok,,,tn l n-52 (Vs--f + Vsingkn) n

I tkl2 ~iln~ r=2 nl (3.3.93)
0sine o 1 nI sin2 10-ek(.l fo2 -il< tl)n
2

From (3.2.7), (3.2.8), and (3.2.11),

(3.3.94) 1 S31 = (silO for -1 ! t, (1.
(n2 ) 2

We use estimates (3.2.7), (3.2.8), and (3.2.11), to write

IS41 = 0 n3 t-, xl) (3.3.95)
o sine
n3 sin le - eOk.l "
2

From (3.2.7), (3.2.8), (3.2.10), and (3.2.11),

IS51 = 0( xl)
(3.3.96)
0 sine
n3sin2 I -(,.
2

72

Using (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2), we have

Vgsinak, n (Vsin + Vsink, n) Is I ( nO tk,. - x12

(3.3.97)

0( sine ) = 0 n~ in2 18 7k,=
2

for -I <

2

Estimates (3.2.8) and (3.2.11) imply that

IS61 0 (sin6 fOor

The lemma now follows. Lemma 3.8

Ix - tk,.J Irk., (X) I = 0

1 k < 1.
tk2

(3.3.99)

for IOkDn - O < C,k = 2, 3, ..., n-1.
n

Proof:

We begin by showing that rk,,(X) can be written

in the form

rk., (x) =

(3.3.100)

(I - x2) P12 1 (x) (I - 2.n) (X - tk,) 2 p'_ )k"

(1 - x2) P1_1 (x)
tk.,) 2[p (tkn)]3 m [l. n (n - ) - r (r - I r-2 n(nf-) + r(r -)J

* (2r - 1) P-1 (tk ) p'(;)

(3.3.98)

(sin6\
n/

Notice that from (2.2.4) and (2.2.10),

E [ n (
] n(n - 1) - r(r - i
.r-2 n (n - )+ r (r

* (2r - 1) P,-l .(kn Pf- (x)

n-1
- (2.r - 1) P,-l (tkfl) Pf1 (x)
r-2
+2n(n1)n- (2r - 1)
r=2 n(n - 1) + r(r - 1) P11 (tkn) Pl-1 (x) = (n - 1)

P-lt kx) Pn-2 (tk,)
tk, n - X

(3.3.101)

n-i + 2n(n - )
z=2

+ Pn-1 (x) P.-2 ( tkl)
( tkn - X) 2 (2r - 1)

n (n - 1) + r(r - 1)

( - ,n) Pa1 (tk,.) P _ (X)
tkn - X

(1 - tk,) Pn-l (tk,,) Pn-l (X)
(tkn - x) 2

1 n-- (2r + 2n(n - I
z=2 n(n - )+ r (r -

' P,-l tk.,n) P'Pl1 iX)

ï¿½pI_1 (tk, n) P_1 (X).

(1 - X') .PL2 (X)

[Pl_,i ( tk, n) ] 2

(3.3.102)

(1 - x2) P1 (x) (1_t2 )2 [pl/

1( - t.n) P1 (tkl) P-.1 (x)
Ik, n - X

+ (1 - P-1 (tk, n) P,-1 (X)
( tk,X - 2

+ 2n (n - 1)

n-I
z-2

(2r-i) P /
n(n 1) + r -(r -) 11(tk,)P1 (X)

(1 - x2) P_1 (x) P"_, (x)
(1 - t2,n) (X - tk, ) [PA-_ (tk,) ]2

2n(n - 1) (1 - x2) Pn-1 (x)
(1 -Pn- (tk.n)

(2r - 1) P,-1 (tk,n) PI-1 (X)
n(n - 1) + r(r - 1)

From (2.1.12), we see (3.3.100) is correct.

calculate the order of Ix- tk,nl Irk,f (x) I.

Next, we Recalling

(3.2.2) - (3.2.5), (3.2.7), and (3.2.19), we state

Hence,

rk.n (x) =

n-I
r-2

(1i-tk,=)(X - t,") 2

(1 - x2) P1 _i (x)
(1 -tk, .) I X - tk, , I ( p.' , ) ], 2

(3.3.103)

= (1 - X2) P"-I (x)
(1- tk, n) P;,-1 (tk, n)

= 0( sinO yrsin-ff

From (3.2.2), (3.2.3), and 1 - ek,nl < c we have
n

sinO
sink,n

+ sinf - sinek,n
sinek0 .

(3.3.104)

=1 +

2 cos 0 + ek, sin lo - ek.=l
2 2
sink, n

2sin le - ek.nl
1+ 2 =0i.
sinJc,n

From (3.3.103) and (3.3.104),

(3.3.105)

(I - x2) P2-, (X)
1 t ) 1 x - tkn [Pn-l (tk,n) ]2

O(sine)
nT)

Recalling that le - 8k,n I < c, we have
n

Ix - tknl sin eke. sinI-ek,.l
2 2

(3 3 106) [sinocos ek,.o-BI + fcoselsin Ik,.ï¿½-f ]l (3316 ï¿½ 2 2 In

o (sin +0 1

It follows from (3.2.7), (3.2.9), and (3.2.11) that

(1 - x2) lP-1 (x) I
(1 - t2) 2i p1 I3 Pk-n (k, n)

(3.3.107) n-1
r-2 n(n + r(r-i)

* (2r - 1) IPrl (tk,n) PT-1 (x) I : 0(1).

Using (3.2.7), (3.2.8), (3.2.10), and (3.2.11), we obtain an alternative estimate.

(3.3.

108)

From (

(1 - X2) 1P _l (X) I
( 2 - 21.n)lIP (tkn) 13
n-1 tn
z [ n](
r-2 n (n - I)+ r~r I

* (2r - 1) I P,1 (tk, n) PI-1 (x) I = 0(nsin).

3.3.105) - (3.3.108), the lemma follows.

Lemma 3.9

(3.3.109)

1(-x) Z,, (x)I=O ( Sine) fo 1 < x <1

Proof: Using (3.2.7),

(3.3.110)

Define

(3.3.111)

k(x) = (1 - x) (1 - X2) p"_1 (x)

n-2 (2r - i) P '-1 (x)
r=2 n(n - 1) + r(r - 1)

+ (n- 2) ( x) (1 x2)pn_(x) (
2( n 1 ) w e- s e e t h a t i

Recalling (2.1.13), we see that it is left to show

(3.3.112)

As in Lemma 3.7, define

(3.3.113) ar,, = n(n - 1) + r(r - I) .

(1-x2) IPI-1 (x)P-2 (x) = 0(sin)
2 (n)

jk(x)l = 0 (sinl).
n

We next apply (3.2.1) to the following

(2r 1) P-j (x)
n(n - i) + z(r - 1)

z=2 ar,n

s=2 n-2
s-2

(3.3.114)

(2s - 1) Ps-1 (x) + (2s - 1) P;-, (x) =

n(n - 1) + (n - 2) (n - 3)

n-3 2 X 2.r r= r,n r+l,n

I /
* (2s - 1) P '1 (x) P"-1 (1) +
s=2

n-2/
T, (2 S 1) P-1 (x) P'-I (1)
s-2

From (2.2.8) and (2.2.10),

r
E (2s - 1) P..11 (x) P,,-, (1)

(3.3.115)

-- /[P - (X)z -x r' (X)

Pr-1 (X) - Pr (X)
1 -x

1
2 (n2 - 3n + 3)

+ P'-j (x) - P, (x)
(1 - x)2
/X 2

+ P'i (x) + Pr (X)
1 -X

+ 1) '- (x) - i (x) +
(1 -x)

Hence,

(3.3.116) k(x) = Al + A2 + A3 + A4 + A5

where

1]
dr+l,n

2Pr (x)
(1 - x)

A, = (1 - x2) P, (x)

(3.3.117)

n-3 2r(r + 1) / E 1 [P-1 (x) Pr (X) ] .r-2 (r,n r+l,n

A2 = (1 - x2) P_1 (x)

(3.3.118)

(3.3.119)

n-3 4r I (x)
r=2 a ar,nar I,n

-1 1) (1 -x2) A3 = 2 (n2 - 3n + 3)

.P,- (X) [P.3 (x) - Pn2(x) I

(3.3.120)

A4 = 1 (1 - x2) P"_1
(n2 - 3n + 3)

(x) Pa-2 (x) I

and

(3.3.121)

(n -2) X) (1 X2)
A5 = 2(n 1) - ( xPn1(X P)-2 (x)

From (3.2.8) and (3.2.10),

(3.3.122)

IA21 = O sin0 n = O si_)
x-2 n 4 sinO n

and

(3.3.123)

I A41I = 0 ( )sinl)

Using (3.2.1) and the observation that we have a telescoping series,

M r(r + 1) [ ) I
r-2 rn r+l,n.

= 4_(_+1) _ (r+ 1)(r+2)
z-2L ar,n r+l,n (ar+l,nar+2,n

(3.3.124)

I I z (n - 3) (n - 2) E [P'_1 (x) - P, (x)] +
s-2 -n-3, n an-2, n

n-3
F, [P.1 (x) -Ps(X)]
s=2

n-4 2(r 1) [n(n -1) - r(r + 2)]
r-2 a1, n r+l,n zt+2,n

1 [1 - P(X) ] + (n - 3) (n - 2) [l x .en-3,n an-2,n

Together (3.2.8) and (3.2.10) along with (3.3.117) and (3.3.124) imply

(3.3.125)

( n-4
IAl 1 0 sin8 T r
z-2 n 3 sinO

+ sinO)

= 0(_21110).

Lastly,

(3.3.126)

2 (n2 - 3n) ( - x2) P3-) (x) [Pn-3 (x) - P.-2 (x) + (1 - x) P1-2 (x)]

+ (-2n2 + 6n - 5) (1 -X2) 1
2 (n - 1)2 (n2 - 3n + 3) (1 - X) P,1 (X) P- (x)

The estimates (3.2.8) and (3.2.10) imply that

(3.3.127)

(-2n2 + 6n - 5)
2 (n - 1)2 (n2 - 3n + 3)

(1 - x2) (1 - x) P'-j (x)

PI-2x W=O

Using (2.2.5), we obtain

P.I3 (x) - PI2 (x) + (1 - x) Pp-2 (x)

(3.3.128)

= P,13 (x) - xP_ 2 (x)

= -(n - 2) P_2 (x).

Hence, from (3.3.128) and estimate (3.2.7),

(3.3.129)

(( - X2)
2 (n2 - 3n + 3)

P", (x) [P=-3 (x) - P-2 (x)

+ - X 2 (X)] = 0 (1ï¿½ We conclude that

(3.3.130)

JA3 + A51 = 0 (sJfl)

and the proof is completed. Lemma 3.10

(3.3.131)

nI= tk(sIe) for)I1= o xi!5) k-0 n o- 1

Proof: We begin by noting that we need only prove the result for -1 < x < 1 since (1 + x)ro,n(x) and (1 - x)r,(x) vanish at x = 1 1, and rk,n (ï¿½1) = 0 for k = 1, 2, ..., n - i.

We now look at the terms i + xI Iroo(x) I and

1 - xl Irn.. (x) I. From Lemma 3.9,

(3.3.132)

I' - X1l jr.'.(X) I = 0 (sine) for -1 < X (1i

Since rn,n(-x) = ro,(x) from (2.1.13), we have

(3.3.133)

j+ XlI -r-, (X) I = I' - (-X) I I-rn,n(-X) I

0 (sine) for -1 < X < 1

Hence, it is left to show that

(3.3.134)

n-1
E k -k X fxl

Actually, we shall see that we need only prove the above inequality for -1 x < 0.

It follows from symmetry and uniqueness of the rk, (x)'s, that

(3.3.135) rkf,(-x)=zl-k.,(x) and tk,, = - k,, for k = 1, 2, ... n -I

Further,

n-1
I 1 (-x) - tkl In rk,n (-x)
k-I

(3.3.136)

n-1 = - Ix - t.-k J I -r.kf (X) I
k-I

E 2 Ix X tknlJ trkfl(X) k-1

Therefore, proving (3.3.134) for -1 < x : 0 is sufficient. To begin,

(3.3.137)

n-1
M I - tknl Ik. .(x) I k-I

= z X - tkoo Irk., (X)I

189k.nI < -~
n

+
i8~k.nI ~

Ix - tk,.l Irk,. (x) I

From (3.2.6), there are a finite number of different Ok,.'s such that 1 - ek,.nI < c. From Lemma 3.8, we may
n

conclude that

(3.3.138)

<
le-ek ,, < S
ï¿½ n

IX - tk,I Irk (X) I=0 (sinO

From Lemma 3.7,

I X -tknl -k, (X)
0-k.

(3.3.139)

0sineO {f2jl iSkI
IOn 2 nsin L _ -___ ,neo., 2

+ 1 +1
n 3sinl3 6-k..I n
2 JJ

After applying (3.2.17) and (3.2.18) to (3.3.139), we

have

(3.3.140)

JX tknl Irk,nx W 0 (si.ne

I88k.nI ~
11

The proof of the lemma is now complete. Lemma 3.11

For -1 < x . 1,

(3.3.141)

n
n Irk )I = 0(X)
k-0

Proof: From the definition of rkf(x) in (2.1.13) and the estimates (3.2.7), (3.2.8), (3.2.9), and (3.2.11), we have (3.3.142) ir=,,(x) I = II(-x) I=)(1) for -1

n
Note that because Z Irk, n(ï¿½I) = 1, it is sufficient to k=0
prove (3.3.141) on the open interval (-1,1).

Following an argument similar to (3.3.136), we have

(3.3.143)

n-1 n-1 1- I rk. (x) I = T I rk. ,(x) k-i k-i

We are, therefore, left with the task of showing

(3.3.144)

n-i
1 jrk,, (X) f =Q fOr
k=1

-1 < x < 0.

From Lemma 3.8 (3.3.100), we may write

(3.3.145)

rk,. (x) = Ck,. - Dk,.

where

(1 x2) P'_21 (x)
(1 t k,n) (X - tl,n)[Pfn- (tk, n)

(1 - x2) p'_1 (x)

ï¿½-n-111
- r - (n 1) + n 1)

(2r - 1) Pr1 (tkl) P i (X)

Ckf n=

and

Dkfn =

t . ) 2 [ P - t . ) ] 3

A result of Turin [11] states that

n-1 (1 - x2) P 2_. (x)
I~ ~~ (I-t) iX - tk. ) 2 [p"_l ( tk, ) ] 2
(3.3.146) k' (1- ( t [ t

= 1- Pt1(x) = (1)

Hence, we need only show that

n-I

(3.3.147) E= 0 (1) for -1< x 0,
k--l

and the lemma is proved. We now restrict x to the interval

[-1,1].

From (3.3.107),

(3.3.148) IDknI= C) (1)

We proceed to obtain an alternative estimate of Dk,. Recall that we earlier defined

(3.3.149) ,r, = n(n - 1) + r(r - 1)

From summation by parts (3.2.1) and (2.2.10), we obtain

(U - x2) P.1- (x)
Dk,. (1 - ")2(pn_ t ,

2 [a nln I)ï¿½ (z n(n 11 (x +i (3.3.150) -(r- n(n +r(ri12S -1) P._ (tkn)P/.1(X) + n(n - 1) - (n - 1) (n - 2) &-2 -1n(n I) + (n -I) (n -2)

n-I 1 }
E (2s- - ) P -1 (tk) P'_1 (X) = s .k + s2. + s3. + SI.k
s-2

4n(n - 1) (1 - xl) P,-1 (x) ( - t ) [p1"_ (tk.f)] (tk] f 3 X)

n-2 r2 .r=2 O r,n n r+l,n r-(X ,(t,) -P ;) I1(tkn

4n (n - 1) (1 - x2) Pn-. (x)
(1 - t,) [p, (tk.) ] 3 ( tkf. - x) 2

n-2 2 . a, [Pr-i (X) Pr (tkfn) - P, (X) Pr-l (tkfn)] r=2 ar,nar+l,n

-(1 - x2) P-. (x)
( k . ) [pn_l ( tk .= 3

-(1 - X2) P-1 (X)
(1 tk,,)2 [p,_l (tk,,) ]3

tk, n - X

p_ k (X) n_2 (t.n)
( tk, n - X) 2

where

S,k =

S2,k -

S3, k= \$4,k x

From (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1),

o sine n-2 IS2,.kI = 0 t
tkn -x Ir-2

r(sine + sinekn)
n4

(3.3.151)

( 1 1
0 n2sin2 16-5,1 for -i < tk,n .
2

We use estimates (3.2.7), (3.2.8), and (3.2.11) to get

(3.3.152)

IS2,, = 0 ) for -1 t. < 1

From (3.2.7), (3.2.9), (3.2.11), and (3.2.16),

(3.3.153)

Iksin0,n0)
IS3.kl = 0 (n 21 k,.-- -XI

-0 (n2sin2 le- k.1)

Using (3.2.8), (3.2.11), and (3.2.16), we obtain

(3.3.154)

1S4,kI 0 ( sin2 s Okn 02 12 in22
'la, l~~~~ = [,-F- 0 ( 00ko

Finally, we estimate the order of S1,k. From (3.2.1)

and Lemma 3.3,

4n(n - 1) (1 - x2) P.-1 (x)
( - t,)[P1(tko)]3 (tkn _ X)

r=2 L z,nar+l,n

: [P -i (X) Ps (tk,.) - P8 (x) P.- (t,..) ] s-2

+ (n - 2 )2_ n-2 + Ex [-1 (X) P, (tk,)
an_2,n(nSl,n S,2

4n(n - 1) (1 - x2) Pn-_ (x)
(1 - tk.1) [P- (tk.l) 33 (tk,f - X)

(3.3.155)

r=2

(2r + 1) [n(n - I) - r(r + 1)]
ar,n r+l,n cz+2,n

r(P ,.. (x) -Pr(X) P ( tk.) + 1 - P ' (x) P (tk.l) Pr-1 (x) P ( tk.,) - PI(X) P-, ( tkn) (n 2)2
tkn - X I n-2,n n-1,n

(n - 2) (P-3 (X) - P-2 (x) ) P-2 (tkfl)
1 1 -x

Pn-2 (x)P-22(tkfl) - (n - 2)

P-3 (X) Pn-2 (tkn) Pn-2 (X) Pn-3 (tk,)
tk, n -X

Sl,k =

(r + 1)2 ar+l,nar+2,n

- P.' (x) P.-1 ( tk ,.)]}

We use estimates (3.2.7) and (3.2.11) to get

4n(n - 1) (. - x2) P-12 (x)
(1 tk.,) 2 [P-1 (tkl) ]3 (tk" - x)

M f3 (2r + 1) [n(n -1) - r(r +1I
r-2 tzr,n +l,n ar 2,n

r(P, (X)-P(x)) P(tk ) + 1

+ ( n - 2)2
Ccn-2,n dn-l,n

(n -2) (P-3 (x)
1 -x

o0 sin O
nx tk I

P 22, (x))

P.-2 (tk.n) +

0 ~n2 sin2 le-e,.l
2

From (3.2.7), (3.2.9), and (3.2.11),

4n(n - 1) (I - X2) Pn-3 (X)
1 (1 kna [PI-1 (tk,,,) ] (tk, X

(3.3.157)

r=2

(2r + 1) [n(n - 1) - rr + 1)]
(Mr,na(x+I, n (c+2, n

P p(x) P, (tkn) +

-0 2 o -I x )

- 2)2 P'-2 (X) Pn-2 (tk)
aZn-2, na{n-l, n

0( 1 )
sin2 le-e,,,
221f

(3.3.156)

We next use (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1) to

get

4n(n - 1) (1 - x2) P.-I (x)
(I tk.) 2[P.-, (tk")]3 (tkn - x) 2

E3 (2r + 1) r [n(n-I) - r(r+1)
Xr-2 ar, n 1,n r+2,n

[P1-I (x) Pz (t:k..) - Pr (x) P -1 (tkf)]

(3.3.158)

+ (_- 2)3 [P.-3 (x) P.-2 (tk.)
nm-2.n n-1, n

-P.-2 (X) P.-3 (tk,.) ] }

sin (sinO + sinok,.)
n( 21tk,. - X12

0 ( n I1 - for -1 < tk~n .
In Sjnz l I-ek..I 4 2
2

From (3.2.7) and (3.2.11),

4n(n - 1) (1 - x2) P--I (X)
(1 tJ. ) [P3 (tkkn)]( t -x)2

E 3 (2 + 1) r [n(n- 1) -r(r +1)]
|r-2 Cgr, nOcx+l,ndez+2,n

- [P-1 (x) P1 (tk,.) - P7 (x) P,_ (tk,.)

-2) [P.-3 (X) P.-2 (tkf.) =.-2 (X) .n tk.n I.
0 -)for -1 ! tkn, < 1.

(3.3.159)

It follows from (3.3.155) - (3.3.159) that

(3.3.160)

0 c- 1 (+ 1 for
n n2sin 2 l8ek. for - n n
2

Hence,

(3.3.161)

I~,[ + 1 IDI= n nsin2 16 - n.ij
2

We now break into two different sums as follows.

(3.3.162)

n-1
k= I Dk

From (3.2.6), the first sum on the right hand side of (3.3.162) contains a finite number of terms. Hence, (3.3.147) implies that

(3.3.163)

E ID.,I = 0 (1). le-ek,.lI<-0

From (3.3.161) and (3.2.17),

(3.3.164)

Dk.-I = E 1 +1 --I - I = 0 (1).
18e-1: n n2 i 2 ) ,

The proof of the lemma is now complete.

IDkl +

IDk,. I

l E-e , -

C n

Proof of Convergence Results

We begin this section with a result of DeVore, Theorem

2.4 in [10]. Let Lip 1 denote the class of Lipschitz one functions and C[-1,1] the class of continuous functions on the interval [-1,1]. Theorem 3.A

Suppose that L, is a bounded linear operator on C[1,1]. If C1 > 1 is such that for each g E Lip 1, we have (3.4.1) Irn(g;x) - g(x) I C1 JIII for -1 X

n

Then for each f E C[-1,1], we have

(3.4.2) ,L..(f;x) - f(x) I C2 (f; I II) for -1 5 X )i

where C2 is independent of f, x, and n.

The preceding is a specific case of the more general theorem stated by DeVore.

We next make the following observation which follows from the uniqueness of the rk,.(x)'s.

n
(3.4.3) E rkan(x)- 1
k-0

Proof of Theorem 3.1

Let f E Lip 1. Then using (3.4.3) we have

nn
R, (f ;x) - f (x) E= f- f( tk, n) rk, n (x) f f(X)

(3.4.4) = E [f(tk,,) - f(x)] rk, f(x)
Ik-a

n
E Ix - tk,J Irk, n(x)
k-0

for some absolute constant 1. Hence, using Lemma 3.10,

IRn(f ;x) - f(x) = T I X - tkl- rkf(x) i) (3.4.5)
= 0 for -1 < x < 1.

From Lemma 3.11, the operator R is bounded. Hence, the theorem follows from Theorem 3.A.

Full Text

PAGE 1

ON SOME PROBLEMS OF INTERPOLATION AND APPROXIMATION THEORY By JOHN BURKETT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 UffiVaSJTY OF FLORIDA LIBRARIES

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To my Parents Frederick and Catherine Burkett

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ACKNOWLEDGMENTS The author wishes to express sincere thanks to Dr. Arun Kumar Varma for his many hours of discussions and counseling in the field of approximation theory. His contagious enthusiasm for the subject and his desire to share this enthusiasm made this dissertation possible. The author is grateful for the generous advice and comments made by the members of the Ph.D. committee. They are Dr. Edwards, Dr. Mair, Dr. Popov, and Dr. Sheppard, in addition to Dr. Varma. Special thanks belong to Dr. Szabodos for his kindness and suggestions during a visit to the University of Florida. Finally, the author thanks Dr. Drake and the Department of Mathematics for their support. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii ABSTRACT v CHAPTERS One INTRODUCTION 1 Polynomial Approximation 1 Lagrange and Hermite-Fe jer Interpolation .... 5 Birkhoff Interpolation 6 Lacunary Spline Interpolation 15 Markov Type Inequalities 19 Two EXPLICIT REPRESENTATION OF A ( 0 ; 2 ) PROCESS ... 24 Introduction and Main Results 24 Preliminaries 27 Proof of Theorem 2.1 28 Three CONVERGENCE RESULTS FOR A (0;2) PROCESS 33 Introduction and Main Results 33 Preliminaries 34 Lemmas 37 Proof of Convergence Results 93 Four LACUNARY INTERPOLATION BY SPLINES 95 Introduction and Main Results 95 Proof of Theorems 99 Five EXTREMAL PROPERTIES FOR THE DERIVATIVES OF ALGEBRAIC POLYNOMIALS 110 Main Results 110 Lemmas 112 Proof of Theorems 122 Six SUMMARY AND CONCLUSIONS 131 REFERENCES BIOGRAPHICAL SKETCH 138 IV

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE PROBLEMS OF INTERPOLATION AND APPROXIMATION THEORY By JOHN BURKETT December 1992 Chairman: Dr. Arun K. Varma Major Department: Mathematics We start by presenting some well known results concerning polynomial approximation, Birkhoff interpolation, lacunary spline interpolation, and Markov type inequalities. These results provide a historical motivation for the problems considered in later chapters. After this introduction, we give the explicit representation and convergence properties for a Birkhoff interpolation process. We find the unique algebraic polynomial that takes on function values for a specific set of knots. In addition, the second derivative of this polynomial takes the value zero on another set of prescribed knots. Next, we look at how well these polynomials approximate a given function. It is shown that the polynomials converge uniformly for any continuous function on the closed interval. In fact, we present a pointwise v

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estimate which provides a discrete, interpolatory proof of the Teljakovskii Theorem of Approximation. Next, motivated by earlier results of Meir and Sharma, we consider lacunary spline interpolation in the (0,1,3) and (0,1, 2, 4) cases. For (0,1,3) interpolation, function values and third derivative values are prescribed at the joints, while function values and first derivative values are prescribed at the midpoints of the joints. Similarly, we consider the problem of (0,1, 2, 4) interpolation. One such spline turns out to be local in character. Specifically, it is determined by the solution of a diagonal matrix. Motivated by an open problem of P. Turan, Rahman studied the extremal properties of polynomials under curved majorants in the uniform norm. We discuss similar results in the L p norm. We present theorems for both the circular and parabolic majorants. vx

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CHAPTER ONE INTRODUCTION Polynomial Approximation In 1885, Weierstrass showed that an arbitrary continuous function on a compact interval can be uniformly approximated by a sequence of polynomials. We more precisely state this as follows. Let f g C[-l, 1] . Here C[-l,l] denotes the class of functions continuous on [-1,1]. For any e > 0, there exists a polynomial P(x) such that ||f(x) P(x)|| < e. Here, we denote the usual sup norm, = -Si ifÂ«iIn 1909, Dunham Jackson chose, as a thesis topic, to investigate the degree of approximation with which a given continuous .function can be represented by a polynomial of given degree [21]. This gave rise to the Jackson Theorem. Before presenting the theorem, we define the concept of best approximation for a given function. Let f e C[-l,l]. Then (1.1.1) EJf) pj* |f pj 1

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where n n represents the class of algebraic polynomials of degree ^ n. Theorem 1 . 1 (Jackson). Let f e C[-l,l]. There exists a positive constant A such that (1.1.2) EJf) zAulf,Â±) for n = 1, 2 , ... n where A is independent of f. Note : ' |x?yf<8 l f <*> f M\A more rapid decrease to zero for E n (f) is possible if we assume more smoothness for f. Dunham Jackson proved the following result on differentiable functions. Let C r [-l,l] denote the class of functions f such that f (r) 6 C[-l,l] . Theorem 1.2 (Jackson). If f e C r [-l,l], then (1.1.3) E n {f) <. A ( Â— ) r 0 ) (f iz) , Â— ) for n = 1, 2, ... n n where A,, is a constant independent of f. In 1951, A.F. Timan obtained the following improvement of the Jackson Theorem. Theorem Â— 1 . 3 (Timan). Let f 6 C[ Â— 1,1]. There exists a positive constant B and a polynomial P n of degree n such that

PAGE 9

3 (1.1.4) | f(x) P n (x) \ Â£ B 0) HE* + Jf, \ n J \ n 2 )\ fÂ° r -1 ^ x ^ 1 and n 1, 2, ... where B is independent of f . Tinian's theorem gives a pointwise estimate for |f(x) P n (x) | . Notice that as |x| approaches 1, the order of convergence near the ends is better than at the middle of [1 , 1 ]. We now state a needed definition. Let Lip a represent the class of functions f in C[-l,l] such that (1.1.5) | fix) f(y ) | ^ M\x-y\ a for all x and y in [-1, 1] where M is some fixed constant. We now state a converse to the Timan Theorem proved by V.K. Dzjadyk in 1956. Â» Theorem 1.4 (Dzjadyk). Let f e C[-l,l] and 0 < a < 1. There exists a constant B and a polynomial P n of degree n such that (1.1.6) |.f(x) P n ix) | <. B \ 1 )Â“ l n ) U 2 J and n 1, 2, ... if and only if w(f,h) <; ChÂ« for some constant C.

PAGE 10

4 A further improvement of the Jackson and Timan Theorems was made by Teljakovskii in 1966. Here, it was shown that the estimate of |f(x) P n (x) | can be made exact at the endpoints of the interval. Theorem 1 . 5 (Teljakovskii). Let f e C [-1, 1], There exists a positive constant D and a polynomial P n of degree n such that n = 1, 2, ... where D is independent of f. Modifying the Jackson operator, Ron DeVore strengthened the above theorem as follows [10]. For a function f(x) bounded on [-1,1], we define the second modulus of smoothness by Theorem 1.6 (DeVore). Let f e C[-l,l], There exists a positive constant A and a polynomial P n of degree n such that n = 1, 2, ... where A is independent of f. There are a number of proofs for the Teljakovskii and DeVore Theorems . The initial proofs involved convolution of the approximated function with the Jackson kernel. This ( prz~2\ (1.1.7) \f{x) -P Ax) | <. Du f, ^ Â— Â— for -1 ^ x Â£ 1 and V n ) ( 1 . 1 . 8 ) 2fisl, \h\ii l fU) ~ 2f ( x+ M U+2h) | . (1.1.9) for -1 Â£ x Â£ 1 and

PAGE 11

5 requires the function to be known almost everywhere. In 1979, Mills and Varma obtained a discrete, weakly interpolatory proof of the Teljakovskii Theorem [24]. In 1989, Varma and Yu obtained such a proof for the DeVore Theorem [48]. These proofs require the function values to be known at only a discrete number of points . This concludes our brief discussion on polynomial approximation. The reader may obtain other important contributions from the book of Timan [39]. Lagrange and Hermite Feier Interpolation Let X denote an infinite triangular matrix with all entries in [-1,1] x, 0,0 (1.2.1) X : Â°.' 1 x i.i X 0.n X l,n x n.n We denote by L n [f,x;X], the Lagrange polynomial of degree s n which interpolates f(x) at the nodes x kjI1 for ^ = 0 , 1, ..., n. Then (1.2.2) L d [Â£,x;X] = Â£ Â£U k , n ) l k>a (x) where k,n (X) = ( X ~ X k.n) <*'nU k , n ) U n (x) = 71 (X~X 1 k=o k,n>

PAGE 12

6 For a time, it was thought that for some matrix X, the Lagrange interpolating polynomials converge uniformly to any given continuous function on [-1,1]. The hopes for this idea vanished when Bernstein and Faber simultaneously discovered in 1914 that for any triangular system of interpolation points , we can construct a continuous function for which the Lagrange interpolatory process carried out on these points cannot converge uniformly to this function. In 1916, L. Fej<Â§r showed that if instead of Lagrange interpolation we consider Hermite-Fe jer interpolation the situation changes [17]. The Hermite-Fe jer polynomials H n [f,x;X] are of degree Â£ 2n + 1 and uniquely determined by the conditions (1.2.3) H n [f,x kn ;X] = f(x ka ) , H' a [f,x k a ;X] = 0 for k = 0, 1, ..., n. Fejir showed that for particular matrices X,. as in (1.2.1), the Hermite-Fe j6r interpolating polynomials converge uniformly to any given function f e C[-l,l] . For example, choosing the knots to be the zeros of the Tchebycheff polynomial T n (x) = cos n9, x = cos9 guarantees convergence for the entire class of continuous functions on the closed interval [-1,1]. Birkhoff Interpolation In problems of Hermite type interpolation, function values and consecutive derivative values are prescribed at

PAGE 13

7 given points. In 1906, G.D. Birkhoff considered those interpolation problems in which the consecutive derivative requirement can be dropped [6]. This more general kind of interpolation is now referred to as Birkhoff (or lacunary) interpolation . The problems of Birkhoff interpolation differ greatly from Lagrange and Hermite interpolation. For example, Lagrange and Hermite interpolation problems are always uniquely solvable for a given set of knots, but a given problem in Birkhoff interpolation may not have a unique solution. More precisely, given n + 1 integer pairs (i,k) corresponding to n + 1 real numbers y ljc and m distinct real numbers x*, i = 1, 2, . .., m Â£ n + 1, a given problem of polynomial interpolation is to satisfy the n + 1 equations (1.3.1) P n ( *> (x i ) = y ik with a polynomial P n of degree at most n. (We use the convention P n to) (x) = P n (x).) For each i, the orders k of the derivatives in (1.3.1) form a sequence k=0, 1, ..., ki. If one or more of the sequences is broken, we have Birkhoff interpolation . A number of different cases in Birkhoff interpolation have been studied. In its first general treatment, Tur&n and associates studied (0,2) interpolation where the knots

PAGE 14

are the zeros of the integral of the Legendre polynomial [3] [4] [34]. It was found that these interpolating polynomials exist uniquely only when the number of knots used is even. We state this result as a theorem. Define (1.3.2) 7l n (x) = (l-x 2 ) p'a-i(x) where P n _ x (x) is the Legendre polynomial of degree n 1 normalized by P n _i(l) =1. An equivalent definition of n n ( is X (1.3.3) x n (x) = -n(n-l) f P fl _ 1 (t)dt. Theorem 1 . 7 (Turan and Suranyi) . Given arbitrary real values (a lfn/ a 2#n/ ..., a n , n ) and (b lfB , b 2#n , ..., b n
PAGE 15

9 Later, Varma and Prasad proved the following [47]. Theor em 1 . 8 (Varma and Prasad) . Given arbitrary real values ( c i, n / c 2/n , ... , c nn ) and (d 2n , d 3n/ ... , d n _ 1>n ) where n is an even positive integer, there exists a unique real algebraic polynomial Q n (x) of degree <; 2n 3 such that (1.3.5) Q a ( x i.n) = c i. a for i = 1, 2, ..., n and Qn ( x i,n) ~ ^ i.n f f Â— 2 , 3, Â« Â• Â• , n Â— 1 where -1 = x lfB < x 2/n < . . . < x n , n = 1 are the zeros of (1 x 2 ) P n . 2 (x) . After answering the questions of existence and uniqueness, it is natural to address the problem of convergence. Tur&n and Balazs followed Theorem 1.6 with a result on convergence which was subsequently improved by Freud [18]. Theorem Â— 1_. 9 (Tur&n and Balazs, improved by Freud) . Let f G C[~l/1] such that
PAGE 16

10 (1.3.7) R a (f.x) = 2 Jc=l f(* k .n) *k , a U) + 2 Pj^Pj^U) JcÂ— 1 where the explicit forms of r k/n (x) and p k , n (x) are given by Balazs and Turan [3]. Other cases of (0,2) interpolation have also been studied. For example, Varma studied the convergence properties of the (0,2) interpolating polynomials where the knots are the zeros of Tchebycheff polynomials of the first kind [41], Here, it has been shown by Sur&nyi and Turcin, the polynomials exist uniquely for an even number of knots [34]. Theorem 1.10 (Varma). Let f e C^-l,!] and let f ' e Lip a, a > Â— . 2 If (1.3.8) 6 i,n for i = 2, 3, n + 1 where x 1
PAGE 17

11 Here u i/D (x) and v 1>n (x) are given by Varma [41]. Saxena and Sharma extended (0,2) interpolation to the case of (0,1,3) interpolation [33]. They considered the problems of existence and uniqueness of polynomials which interpolate prescribed function values, first derivative values, and third derivative values at a given set of points . They also obtained convergence results analogous to Theorem 1.9. In 1989, Akhlaghi, Chak, and Sharma addressed the problem of (0,3) interpolation based on the zeros of x n (x) [1]. They found the (0,3) interpolating polynomials to exist uniquely for every n ^ 4. In addition, explicit forms were found for the fundamental polynomials though complicated in nature. Subsequently, Szabodos and Varma [36] found simpler explicit forms for this (0,3) interpolation and, consequently, were able to obtain the following convergence result. Theorem 1.11 (Szabodos and Varma). Let f e C[-l,l]. Then (1.3.11) where
PAGE 18

12 (1.3.12) R a {f,x) = 2 f (x jia ) i jtn (x) where r j>n (x) are the fundamental polynomials of the first kind. Notice that these (0,3) interpolating polynomials converge uniformly for a wider class of functions than in the (0,2) interpolation theorems we presented. In fact, there is an open problem of Turan to find the "most stable" (0,2) interpolation in the following sense [40]. Problem XXXT . Given a matrix X as in (1.2.1) such that each row contains knots where the (0,2) interpolating polynomials exist, find the matrix that will minimize < l 3 13 > -Si Â£ K.Â«<*> I w ^ ere r k,n( x ) &re the fundamental polynomials of the first kind. Since r^fx^J = 1, we cannot hope to do better than (1.3.14) ^ l-Tjc , n (x) | = 0(1) for some matrix x. Although (1.3.14) has not been obtained in the strictest sense, such a result has been recently obtained by the use of two different sets of knots. This type of

PAGE 19

13 interpolation has been referred to as Pal Type interpolation . Akhlaghi and Sharma have studied (0,2) interpolation on two different sets of knots, namely the zeros of ( 1 x 2 ) p n-i( x ) and P n -i(x) [2]. They established that these interpolating polynomials exist uniquely for n even or odd. In addition, some results on explicit forms were obtained which we will not state here. We do, however, present the following. Theorem 1.12 (Akhlaghi and Sharma) . Given arbitrary real values (&i, n , a 2(I1 , Â•Â•Â• / a n,n) and (b 1>n , b 2>n , ... , b n _ 1#n ), there exist unique real algebraic polynomials S n (x) and T n (x) of degree ^ 2n 2 each such that (1.3.15) S n (x in ) = a in for i = 1, 2, ... , n and s n = bÂ±,n for i = If 2, ... , n 1 and Â» (1.3.16) T n (y in ) = b iiR for i = 1, 2, Â— , n 1 and i.n ^ ~ a i,n Â— If 2, ... , n. Here, x lfn are the zeros of (1 x 2 ) Pn_i(x) and y i>n are the zeros of P n _ 1 (x) . Subsequently, Szabodos and Varma proved uniqueness and existence for a modified (0,2) process very similar to the one described in (1.3.15). Their modified (0,2) process differs in that it also prescribes first derivative values

PAGE 20

at the endpoints Â± 1 . Before presenting the convergence results, we define 14 (1.3.17) RJt.x) = 2 /<**,Â„> Jr=l and (1.3.18) R n (f,x) JE f(x k ) r k (x) k=l + o l n (x) + f'(-l) a 2 n (x) We refer to the paper of Szabodos and Varma [35] for explicit forms. The following are their convergence theorems . Theore m 1.13 (Szabodos and Varma). Let f Â€ C[-l,l]. Then (1.3.19) |f(x) RJf,x) | = O L f.iOl n for -1 * x ^ 1 and n = 1 , 2 , ... Theorem Â— 1 Â• 1,4 (Szabodos and Varma) . Let f be a function such that f 6 C[-l,l] . Then (1.3.20) |fU) RJf,x)\ = Ofj^) 2 M (f',1) V n 2 ) Jc-i k fÂ° r ^ x ^ 1 and n = 1, 2, .... a direct consequence of Theorem 1.12 is that (1.3.14) holds. This resolves Turin's Problem XXXI in a slightly different context. For more results on Birkhoff interpolation, we refer the reader to the book of Lorentz [22].

PAGE 21

15 Lacunary Spline Interpolation In the 1970s, 1980s, and 1990s, several papers appeared in which (0,2), (0,3), and (0,1,3) interpolation problems were solved using polynomial splines and piecewise polynomials. We can classify many of these results into three groups . Before proceeding, we define by S ( n r ,Â’ q the class of splines S(x) such that (1.4.1) i) S(x) e C r [0,l] ii) S ( x) is a polynomial of degree q in [x Â± ,x i+1 ], i = 0, 1, ..., n-1 where 0 = x 0 < Xl < ... < x n _ 1 < x n = 1 . In this first group, the data to be interpolated are prescribed at the joints of a spline as well as at the endpoints of the interval. A. Meir and A. Sharma [23] were the first ones to consider the case of (0,2) interpolation on equidistant knots . They have shown that for arbitrary lacunary data [y 1 }^= 0 and {yf}Â” = 0 , there exists a unique (up to the boundary conditions) quint ic spline S n (x) e C 3 [0,1] with joints at Â— (i = 0, 1, . . . , n) such that n ~ ) ~ y i> s ni^r) y'l (n odd). The boundary conditions are Â•S'n (0) y Q and S a (1) . Moreover, if the given values

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16 ^y. i J > an d {y Q i Ya } are the values and the second and third derivative values, respectively, of a function f satisfying f e C 4 [0,1], Meir and Sharma proved the following convergence theorem. Theorem Â— 1_.,15 (Meir and Sharma) . For the unique quint ic spline S n ( x) that interpolates (0,2) data as discussed, we have (1.4.2) ||5 n (r) fW | s 75n*' 3 G)(f< 4 >,A) + 8 n r ~* ||f< 4 >|| n for r = 0,1, 2, 3. Subsequently, S. Demko pointed out that because of the ill-poised nature of the interpolant defined in the preceding, for a given function f e C 6 [0,1], the error 1^ ~ S J where S n interpolates f (as described above), is not of optimal order as a function of mesh length [9]. He further gave justification to this claim. On the other hand, S. Demko also pointed out that the situation changes if, instead of considering (0,2) interpolation by splines, we consider the (0,3) case based on equidistant knots. Consider arbitrary lacunary data {y i }5. 0/ 5 =0 , and O'O' Yn } Â• There exists a unique quintic spline S n (x) g C 3 [0,1] with joints at j. (i = o, 1, . . . , n) such that

PAGE 23

17 = Yi> s'''U) = y'i, s"(0) = y, h O I and s"( 1) = y" The system of equations that uniquely determines S n (x) turns out to be tridiagonal dominant, and consequently, the rate of convergence is of the same order as that of best approximation by quintic C 3 splines, provided the interpolated data corresponds to the function approximated. (y 1 = f{x i ) , y'l' = f"' ( X .) , yfj = f"(x a ) , y'l = f" (x n ) , f e C 3 [0,1]) . The second group of results deals with special piecewise polynomial methods for solving (0,2), (0,2,3), and (0,2,4) problems. We refer to the work of Fawzy [13] [14] [15]. Later, Fawzy and Schumaker [16] defined construction methods for solving the general lacunary interpolation problem. On the positive side, these methods are shown to deliver theÂ’ optimal order of approximation while being relatively easy to construct. One possible defect remarked by them is that their proposed methods produce only piecewise polynomials . We refer here to remark 1 on page 424 [16]. Here, one should note that the data are prescribed at the knots only. ^' or the third group of results dealing with lacunary interpolation by splines, we refer to the papers of A.K. Varma [42] [43], J. Prasad and A.K. Varma [28], Gary

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18 Howell and A.K. Varma [20]. Here, we allow certain data to be prescribed at the midpoints of the joints, in addition to at the joints of the spline. Howell and Varma [20] obtained deficient quartic splines of the class C 2 [0,1] which interpolate lacunary data (function values at the midpoints of the joints, second derivative values at the joints, and function values at the endpoints of the interval). They obtained the following convergence theorem for these splines. Theorem 1.16 (Howell and Varma). Let f G C X [0,1]. Then, for the unique quartic spline S n (x) associated with f and satisfying the above conditions, we have (1.4.3) |S n (r) f (r) U) | <: a) (f a) ,h) , r = 0,1 and 1 = 2,3,4, and * (1.4.4) |S n (r) U) -r (r) (x)| <; B r , 5 h 5 r 0 ^i \f l5) U)|, r = 0,1 and 1=5, where h is the mesh length. The splines in this group are determined by tridiagonal dominant systems. In fact, we now present a case where the spline is actually determined by a diagonal matrix. Prasdd and Varma obtained the first such case. They prescribed function values at the joints and midpoints of the joints, third derivative values at the midpoints of the joints, and

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19 first derivative values at the endpoints of the interval. They proved the following convergence theorem. Theore m 1.17 (Prasdd and Varma) . Let f e C 1 [0,1]. Then, for the unique quintic spline S n (x) associated with f and satisfying the above conditions, we have (1.4.5) |S n (r) (x) (x) | <; (o , r = 0,1,2 and 1=3, 4 , 5 , and (1.4.6) |S n ' r) (x) f (r) (x) | <> B rl h 6 ' 1 \f< 6) (x)\, r = 0,1,2 where h is the mesh length. This concludes our discussion on lacunary spline interpolation . Markov Type Inequalities In 188^, A. A. Markov proved the following. Theorem 1.18 (Markov). If p n ( X ) is a real algebraic polynomial of degree n such that |P n (x) | ^ 1 on the interval -1 ^ x ^ 1, then we have U5 1 ) 1^1 I p'n(x) \ z n 2 .

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20 Later, A. Zygmund [49] proved Theorem 1.19 (Zygmund). If f is a trigonometric polynomial of order n and p ^ 1, then 2lt i ( (1.5.2) i Â£/|/'(8>|Â»d0 2n Hill, Szego and Tamerkin [19] extended this type of inequality to algebraic polynomials on the interval [-1,1] in the form fi 1 'i (1.5.3) f |Pn(x)| p dx p Â£ An 2 f \P n (x) \ p dx l-i > ^-i / where p ^ 1 and A is independent of n and P n (x) . They noted that the problem of obtaining the best constant in (1.5.3) is extremely difficult. Later, B.D. Bojanov [7] proved the following extension of the Markov Inequality. Â» Theor em 1.2 0 (Bojanov) . Let 1 <, p < <Â». Then for every real algebraic polynomial of degree n, we have (1.5.4) 'i > f | P'(x) | p dx Jt P Â£ ' 1 ' f |r'U) | p dx n > -i > where T n (x) = cos n6, x = cos 6. The following problem was raised by P. Turan at a conference held in Varna, Bulgaria (1970). Let (x) ^ 0 for -1 * x * 1 and consider the class P n ^ of all polynomials

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21 P n (x) = 2 a k x k of degree at most n such that |P_(x) I Â£ (x) = 'Jl-x 2 ) and the parabolic majorants ( (x) = 1 x 2 ) . Theor em 1.21 (Rahman) . If P n (x) is an algebraic polynomial of degree n such that |p n (x) | <; -x 2 for -1 s x s 1, then (1 . 5 . 5 ) .Â“ sl if;wii2M. Equality if and only if P n U) = (1 x 2 ) u n _ 2 (x) , u n _ 2 (x) = .?. in ( . fl ~ 1)9 / x = COS0. smo Â» Theorem 1.22 ( Rahman and Watt ) . For given n ;> 3 , let (1.5.6) X r = \ ID = cos|-^Lj, r = 0, 1, . . ., n-2. If P(x) = (1 x 2 )q(x) is a polynomial of degree at most n such that |gU r ) I * 1 for r = 0, 1, ..., n-2, then (1.5.7) |P<Â»| s |tÂ« (1) | for k = 3, 4, ... where xjx) = (1 ~x 2 )T n _ 2 (x), T n _ z (x) = cos (n-2) 0, x = cos0 .

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22 Further, if P(x) is real for real x, then (1.5.8) |P (Jc) (x+iy) | s (1+iy) | for (x,y) e [-1,1] xR and k = 3, 4, For other interesting results, we refer to the works of Rahman and associates [25] [26] [27] [29] [30] [31]. A natural extension of these ideas is to investigate similar problems in the L p norm. We shall state two such results obtained by Varma and associates in the L 2 norm. Theorem 1.23 (Varma) . Let P n+1 (x) by any real algebraic polynomial of degree at most n + 1 such that |P n+1 (x) | Â£ Jl-x 2 for -1 Â£ x Â£ 1, then x = cos0 . Equality if and only if P n+1 (x) = Â± f Q (x) . Theorem Â— 1.24 (Varma, Mills, and Smith). Let P n+ 2 ( x ) bÂ© any real algebraic polynomial of degree at most n + 2 such that \P n . 2 U) | z 1-x 2 for -1 s x s 1, then -l for j = 1, 2 , 3 where f a (x) = (1-x: 2 ) (x) , u n _ Â± U) sin n 8 sin0 ' i (1.5.10) / [P" 2 (x)f dX
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23 where f x (x) = (1 x 2 )T n (x), T n (x) = cos n0, x = cos0 . Equality if and only if P nt2 (x) = Â± fi(x) . For other results in the L 2 norm, we refer to the work of Varma and associates [44] [45] [46].

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CHAPTER TWO EXPLICIT REPRESENTATION OF A ( 0;2) PROCESS Introduction and Main Results Define (2.1.1) -1 = t 0 , n < t 1>n < ..., t n , B = 1 to be the zeros of (1 x 2 )P a _ 1 (x), and (2.1.2) -1 < x 2>n < x 3 , n , ... < x n . 1/n < 1 to be the zeros of P^'fx). Here, P^x) denotes the Legendre polynomial of degree n 1 with normalization (2.1.3) P n . x (1) = l. The following theorem is a direct result of Lemma2 in a paper by Akhlaghi and Sharma [2]. Theorem 2 A Given arbitary values (a 0#nf a 1/D , ..., a n#n ) and (b 2#n , t>3,n/ Â• Â• Â• t bn-i,n) / there exists a unique real algebraic polynomial R^x) of degree 2n 2, such that (2.1.4) Rn (t j/D ) = a j>n for j = 1, 2, ..., n 1, (2.1.5) R n (-1) = a D/n , R n (l) = a n , nf (2.1.6) R" (x jD ) = b j>D for j = 2, 3, ..., n 1. 24

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25 We note that the above theorem places no restriction on n being even or odd. This is in contrast to other similar processes that have been studied. We now present our results on explicit representation. Given arbitrary values (a 0#n , a lfB , . .., a n#n ) and (b 2 , n/ b 3 , a/ Â• Â• Â• / t> n -i,n) / we wish to find the explicit form of the polynomial R^x) of degree <; 2n 2 such that (2.1.4) (2.1.6) hold. For RÂ„(x) we evidently have the form (2.1.7) R d (x) = 2 a kiB r k>n (x) + 2* b kiD p k (x) , Jc = 0 Jc=2 Q-l where the polynomials r k/n (x) and p k>n (x) are the fundamental polynomials of the first and second kind. These polynomials are of degree Â£ 2n 2 and are uniquely determined by the following conditions . ( 2 . 1 . 8 ) ^k.n k = ' t \ _ / 0 for j * k ' 3' a Â‘ ~ l 1 for j = kÂ’ 1/ 2 f . . . f n 1 ^ r k.n( x i,n) r k.n Â“ 0 / (2.1.9) r n.n^j,n) r o,n ( 1 ) Â“ r n.n Â“ 1 / O.n ^a,n ( 0 / r o.n ( x i,c) ~ r n.n ( x i, a ) ~ Â® (2.1.10) p kn (t ja ) = p kn (Â±l) =0, k 2 , 3 , ... / n 1^ (j = l/ 2, . . . , n 1 and i = The following theorem presents representation of these fundamental 0 for i * k 1 for i = kÂ’ 2/ 3 , . . Â• g n 1 ) Â• the explicit polynomials .

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26 Theorem 2 . 1 The fundamental polynomials p k>n (x) and r k>n (x) are given by ( 2 . 1 . 11 ) Pk.n (x) ~ Pa1 (x) (1-Xfc,a) n(n-l) Pi-i (x*. a ) . "s 1 (2r-l) Pf-, (x k a ) Tt r U) r -2 r(r-l) [ji(xi-I) + r(r-l)] ' k = 2, 3, n 1/ (2.1.12) r^U) =A kin (x) B k/n (x) , k = 1, 2, ..., n1, where (1-x 2 ) (x) p' n . i(x) and 2a(n-i) P n _ x (x) Â• V ( 2 r ~l) * r U) r -2 , n(n-l) + r(r-l) = r o.a(-x) = ~ (1 ^ X) P^U) P n _ 2 U) (2.1.13) + (x) 2 , 2 (2r-l) (x) -2 n(n-l) + r(r-l) * p Â»-> <*> *Â« Â« The relative simplicity of these explicit forms is crucial to proving the convergence results in the next chapter .

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27 Preliminaries Here, we list various known results used in the proofs of the next section. The following identities were taken from Chapter Three of Sansone [32], (2.2.1) (1-x 2 ) p"_! (x) -2 x P'.i (x) + n ( 22 -I) p a _! Or) =0 (2.2.2) (1-X 2 ) P"'-! (X) ~ 4 U) + (fl Â— 2 ) (n + 1 ) Pq~\ (x) (2.2.3) x' (x) = -r(r-l) P r _ 1 (x) (2.2.4) ( 1 -x 2 ) p'_i (x) = (r1 ) p r . 2 (x) (r-1) xP r _ x (x) (2.2.5) xP^-i (x) p'_ a (x) = (r1 ) Pr-l (*> (2.2.6) Pr-l (x) xp '_ 2 (x) = (r-l) 1 g (2.2.7) Pr-! <*) Pr-3 (X) = (2r-3) P r -2 (*> (2.2.8) P r-3 ~ P r2 (^ 1 -x Pi-2 (X) + Pi -3 (X) 2 (2s 1) p s _ x (x) P s _ x (y) S=1 (2.2.9) = r P r -1 M Pr ~ P x U) P z . x (y) y-x The above identity is known as the Christoffel formula of summation. Differentiating (2.2.9) once gives the following. E (2s -1) p'. x U) P (y) 3 = 2 ( 2 . 2 . 10 ) = x Pr-l (X) p r (y) pj (X) j> r-1 (y) y-x P t-1 (x) p z (y) p c (x) P r _ t (y) (y-x ) 2 +

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28 From Szego [38], we write (2.2.11) . Â£ P'slU) Ps-Ay) Pr(x) p' r x (y) -P'-.U) p'(y) 8-2 s(s-l) x-y Proof of Theorem 2.1 We first prove (2.1.11). In view of the uniqueness theorem, it is enough to verify that p k , n (x), as stated in (2.1.11), indeed satisfies the conditions in (2.1.10). The first condition clearly holds. Fix any k e {2, 3, ..., n 1}. From (2.2.1) and (2.2.3) , (2.3.1) ^2 [ P n-lM =Pn-l(Xj.n) *Z (*,.Â»> + PH-ilxj'J n z (x jtn ) =-r(r1) p'. x (x jiB ) n(n-l) P n _ x { Xj J P r _ x ( Xj a ) . Hence, (2.3.2) Pi:,n ^ x j,n) _ Pn-l( x j,n') n(n-l) Pj 2 -Â± (x kin ) . V (2r-l) PUlx t . n ) PU (x, J r -2 r(r-l)

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29 From (2.2.11), we now have Pi, a (Xj a ) = d~ x k,n) P n-l ( x k,n) P a-2^ x k,n^ P a-l^ x j,n^ {2.3.3) n(n-l) 2 {x ka ) . P n-1 ( x i, n ) ^ x j,i 1 ~ x k,a } P a-1 ( x i,n) From (2.2.1) and (2.2.5), (2.3.4) jt Pk, ) P n-l^ x j,n) a j a ~p Tx ) vn-l ' x k,n ) P'n 1 Uj.n) ( X j , d x k,n^ Pa1 ( x k.n) Recalling the fundamental functions of Lagrange interpolation, we see that (2.3.5) Pk.n{Xj, a ) = 0 for j * k and (2.3.6) pÂ£ a {x kn ) = 1 . We conclude, that all conditions in (2.1.10) hold. Next, we turn our attention to (2.1.12). Again, we need only verify the conditions in (2.1.8) hold, and all but the second derivative conditions are clear from (2.1.12). Fix any k Â€ {1, 2, ..., n 1}. From (2.1.12), we have (2.3.7) A kin (x) = U Â° {X) lk ' n (X) (1 -tl n ) Pa-l(t k .) where 1 k,n^ X ) = P n-1 (X) (x-tk.n) p al(t*. a )

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30 Next, we differentiate twice to get (2.3.8) > _ 2n (n l) Pn-i lk, a (Xj ) (1 -tl u ) P' Q lU kin ) = 2n(n-l) [-Pfl-l ( tk.n) ] ( X j,n ~ tjc.fl) 2 Now, from (2.3.1), we have a ) 2 Â‘ (Xj n ) A kiD (.Xj n ) Bk.a (Xj D ) 0 // for j = 2, 3, . . . , n 1 . We conclude that all conditions in (2.1.8) hold. Lastly, we prove (2.1.13). We begin by observing that r o,n( x ) r n,n(~ x ) follows from uniqueness and symmetry of the zeros of P a . x ( x ) and p' a . x ( x ) . We, therefore, need only

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31 verify conditions (2.1.9). Specifically, we show that P n,n (Xj D ) = 0 for j = 2 , 3, . . . , n Â— 1. From (2.3.1), (2.2.10), and finally (2.2.8), d 2 a j? (2r-l) P n-1 ( x ) 7t r ( x ) cbc 2 [r -2 22 ( 22 -I) +r(x-l) A *J,ii = E 2 (2r-l) P x _! (1) (2.3.12) (n-2) ( x j,a> (Xj, D ) P a -2 U 7 , a ) 1 -x. J.n + Pfi-3 ( X j,i ^ Pg-2 [ ( Â”~ 3) P '*2 ~ {n ~ 1] P n-^ X J.J] We now apply (2.3.1) and (2.3.10) to (2.1.13) and obtain (2.3.13) r" (x) d + x i.Â„ ) 2 Â’ [P^-l ( X J.J Pn -2 (*J.n) Pn-l ( x J,n) Pn-2 <*,.Â«)] + P a -, (*j, a ) Pn P n-1 < x i.J [(72-3) P' a . 2 Lx jin ) (72-1 l~ X i.n ~ (n-2) Pn-l^j.n) Pn-iUj.J

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32 Hence , (2.3.14) (1 X 1-J x " (x, ) *n-l (*J.n) a 'Â° j -Â° 2 P B 1 (Xj D ) ^ X ja ^ P n-2 ( x j.a) + P a-1 ( x j,zJ P a-2 i x j,a^ ] + UÂ“-*J.a> P n -2 Uj. a ) + in-3) p'_ a ( Xj n ) (22-1) P'_ 3 (x jiD ) ~ ( 22 2 ) H-XjJ P' a . 2 (X jin ). (2.3.15) From (2.2.1) , * 2P .-1 tXj,.) [ X J.Â„ a ) (22-1) 2 P fl . 2 (x i>a ) (22-1) Pa-3 i x j, a ) Â» From (2.2.6) and (2.2.7), (2-3.17) z''ix. ) =0 ' > p l v \ L n,n u c n 1 KX j.n> Hence, (2.3.18) r" n (x Jin ) = 0 for j = 2, 3, . . . , n 1. This concludes the proof of (2.1.13) and, consequently, the proof of Theorem 2.1.

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CHAPTER THREE CONVERGENCE RESULTS FOR A ( 0 ; 2 ) PROCESS Introduction and Main Results Let f be a real valued function defined on the interval [-1,1]. We now define the linear operator R n (f;x) by (3.1.1) RÂ„n are defined as in (2.1.1). 33

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34 We now present the main objective of this chapter. Theorem 3 . 1 Let f be a continuous function on the interval [-1,1]. Then (3.1.4) | R a (f;x) -f(x)\n ) > d x (n-1) 2 , k Â— 1 , 2 , n1 (3.2.3) (1 'tin) ( n-k ) 2 (n-1) 2 ' k = n-1' 1/ n-1.

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35 (3-2.4) a /k 1/ 2 , Â• Â• Â• / n-1 (3.2.5) |Pi-i(e t , a )| 4 4 --. ~v3 > * ( n-k ) 3/2 .n-1 + 1 , . . . , n 1, (3.2.6) |9 tI -9 w ,J)^. Note (3.2.6) is a direct result of Bruns' Inequality. The following inequalities may be found in Szego [38]. (3.2.7) [PÂ„(x)|-Of 1 ) for -1 \ \Jn sin 8 ) < x < 1 (3.2.8) |P n (x) | = 0(1) for -1 Â£ x * 1 (3.2.9) |p'(x) | = O n ^ sin 3 0 for -1 < x < 1 (3.2.10) |p'(x)| = for -1 < X < 1 From (3.2.2) (3.2.5), (3.2.11) (1-0 2 IO (t*. n ) | 3 o 3/2 n,1c 1,2, Â•Â•Â«, n-1.

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36 From (3.2.9) (3.2.11), (3.2.12) (1 'tin) 2 \PLl(t k>n ) | 3 (1-tiU) 2 l-Pn-lUic.jl 3 (1 \PLl(t kia )\ 6 = o sin 3/z 0 k, n VS / ^ 1/2, . . Â• , xx 1 . The subsequent two results can be found in a paper by Szabodos and Varma [36]. (3.2.13) \P n (x) + P a+1 U) | = O sin 0 N ^ J for -1 Â£ x Â£ (3.2.14) | (x) + P' n+1 {x) | = O n sin 0 for -1 < x ^ From (2.2.8) and (3.2.8), (3.2.15) |P a (x) + P a+ 1 (x) | = O (n) for -1 <: x i Â— . 2 From (2.2.4) and (3.2.9), we may write (3-2.16) | P B . 2 U kiD )\ = O w sin 0 k,n n / h 1, 2, ..., n 1. Finally, from (3.2.2) and (3.2.3), (3.2.17) I 0 0 * Jin 2 sin 2 13 2 = Od) and to 1 1Â— Â» m|h

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37 (3.2.18) From (3.2.19) Lemma 3 . 1 For (3.3.1) and (3.3.2) 2 n 3 sin 3 |9 e. 'lc.nl = O (i) . Erdos [12], P n 1 U) = O (i) Lemmas 1 < x Â£ 1 2 and -1 < y ^ we have r |P r . 1 (x) P r (y) P r (x) P x _ x (y) | o sin 8 + sin \$ k Vsin 0 sin \ \P r -i (x) P r (y) P r (x) P z _ x (y) | = O f V3in 0 + l V* ) where x = cos 0 and y = cos .

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38 Proof ; r |P r _ 1 (x) P r (y) P r (x) P I _ 1 (y) | = r | [P r _, (x) P r (y) + P r (x) P r (y) ] [P r (x) P r (y) * r |P r _ 1 (x) + P r (x) | | P r (y) | + r |p r (y) + P r _ 1 Â» o sin 8 + sin ^ sin 4> sin 0, = Q f sin 6 + sin <|> \ v^sin 0 sin <|> + P r (x) P T . 1 (y) ] | (y) | | P r (x) | In the above, we used estimates (3.2.7) and (3.2.13). To prove (3.3.2), we use estimates (3.2.8) and (3.2.13). |P,-! (x) P r (y) + I P r (y) + P r _ P r (x) p r . 1 (y) | <; |P x _ 1 (x) + P r (x) I |p r (y) | sin 0 . sin Â— + \r) i (y) I I p z I = O N Lemma 3 . 2 For -1 < x ^ Â» 1 2 and 1 < y ^ we have (3.3.3) IP'., (x) P r (y) p'Ax) PÂ„_i (y) | = Q + sin \ Vsin 3 0 sin j and (3.3.4) (x) P r (y) p; (JC) p... (y) | . n C V^ (3in 6 * sln > ) V sin 0 ysin .

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39 Proof : | pU (x) P z ( y ) Pi (x) P r . 1 (y) | = | [pU (x) P r (y) + p' z (x) P z (y) ] [p' z (x) P r (y) + p' (x) P r _, (y)] | * | p'_ x (x) + p' (x) | |P r (y) | + |P r (y) + Pr-l(y) I |PrU) I = O 1 , VsinO sin + sin sin 3 6 ; O i, sinQ + sin(t) ' ^sin^sintj) t Here, we used the estimates (3.2.7), (3.2.9), (3.2.13), and (3.2.14). To prove (3.3.4), we use (3.2.7), (3.2.10), (3.2.13) , and (3.2.15) . | Pr! (x) P z (y) p' z (x) P r . 1 (y) I s I p'_ x (x) + p' (x) | | P r (y) + |P r (y) + -Pr-l(y) I |PrU) I = O Â» Q [ \/r (sin8 + sin) ' V sinQ^sTru}) t r + y/rsin N sin sin0 , Lemma 3 . 3 J 2 [ p -i (x) p * ( t*. a > " Pi (*> P*-i ( t*, a > ] r(P r . 1 (x) P r (x) Â„ , S-X P r^, fl ) + 1 Pr (x) P r ( J T Pr ~ 1 (x) Pr ( tk n) Pr ^ Pl ~ 1 ( tk n ' ) t k.n ~ *

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40 Proof : J 2 [p^
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41 Lemma 3 . 4 Â£ 2s* [P^ (x) P a ( t kiB ) P a (X) P a . k ( t kiD ) ] (3.3.5) = (1 x 2 ) (r + 1) P r (x) ( t k n ) P r -.i (x) P f ( t^) u k,n + p r (x) P^ ( t k n ) P r ^ (x) P c ( t k n ) ( tk.n X) 2 (1 tl B ) (r + 1) P'r ( ^, n ) Pf*l (*) Pr* 1 ( tk.n^ Pr ( x ) X ~ t k-n + Pj*l Pr*l ( ^ k,n ) Pr ( x ) < 2 (r + 1) r[P r (x) P r+1 ( Â£*,Â„) P f+1 (x) P r ( t* >a ) ] + 2 (x t i>a ) Proof : From (2.2.4), we may write J 2 (2s + 1) s [P s _ x (x) P s ( t* >a ) P s (x) P s _, ( ] = S (2s + 1) P s ( t* /a ) [ (1 x 2 ) Pi (x) + sxP s (x) 1 j? 2 + ^ ( X ) [ (1 _ ^Jc.n) -Ps ( Â£jc,.q) + S ^k.n^3 ( ^ Jc,n ) ] (3.3.6) = (1 x 2 ) 2 (2s + 1) P a (t kD ) P' B {x) S~2 (1 J 2 (2s + 1) P s (x) Pi ( t k n ) s= 2 ' + (x t*, n ) 2 (2s + 1) s P s (x) P s U k , n )

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42 We now work with the last sum in (3.3.6). From summation byparts (3.2.1), S (2s + 1) s P a (x) P a (t kin ) (3.3.7) = 2 [s (s + 1)] 2 (2m + 1) P m (x) P m {t kiD ) + r 2 (2m + 1) (x) P a ( t* a ) m-2 r-l E (s + 1) B=2 Pg (x) P s +\ ( Â£^n) -P s -.i (x) P a ( Â‘-Jc.fl X + 2(1+ 3 xt k J + r (r + 1) a = 2 . (x) P r ^ 1 ( n ) -Pf.x (x) -P, ( t k n ) tk.n ~ X r(l + 2xt k n ) . We used (2.2.9) in the second equality of (3.3.7). Now, tk. B ) S (2s + 1) s P s (x) P s ( t^ fl ) S-2 .= i_ 2 s [P s _ x (x) p s ( t kiD ) P s (x) P s _ x ( t*. j ] 2 [P x (X) P 2 ( t kin ) P 2 (X) P x ( t kiB ) ] + (x t*^) (r 2) (1 + 3xt krn ) (3.3.8) r(r + 1) [P r (x) P r+1 ( t kiQ ) P r+1 (x) P r ( ] ~ + 3xt^ /fl ) " j 2 s [P^ (x) P a ( t*, J P a (X) P 3 _ x ( tjC(fl ) ] tjc-a) ( = 3xt* fa ) r(r + l) [P r (x) P r+1 ( t k>a ) P r+1 (x) P r ( tjc>n ) ] .

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Together ( 3 . r 2 s=2 (3.3.9) With further r 2 s= 2 (3.3.10) . 6 ) and (3.3.8) imply that (2s + 1) s [P s _, (x) P s ( t k n ) P s (x) P s _ x ( t ktQ ) ] J 2 S [Ps1 <*) ( tjc. J <*) Ps 1 ( tjt. a ) ] (1 X 2 ) 2 (2s + 1) P s (t k n ) p' s (x) S-2 ' (1 tin) 2 (2S + 1) P s (x) Pg ( t k ) 3=2 ' (1 + 3xt k a ) r(r + 1) [P z (x) P r+1 ( t Ka ) P r+1 (x) P r ( t kiD ) ] 43 calculations on (3.3.9), we obtain 2S S [P_ t
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44 From (2.2.10), (3.3.11) E 2s 2 [P_ x (x) P a ( t k ' B ) P a (x) P s . x ( t kiB ) ] (1 x 2 ) (r + 1) P r (x) P J< , 1 ( n ) P r ^ x (x) P r ( a ) tjc. a ~ * + Â£z_ Pf+1 ( ^k,n) Pr+i ( tic, n) ( fc Jc. a X) 2 (l tÂ£ a ) (r + 1) Pr ( tjt. a ) Pr>l (X) Pf+1 ( Â£jc, n ) Pr X t, Jc,n + Pr ( ^Jc,n) Pr+1 ~ Pr+l ( ^ *,n ) P r (*) <*' t*. a ) 2 + 2 (* t*. a ) " r (r + 1) [P r (x) P rn ( t ifJ1 ) P m (x) P r ( t i>a ) ] The proof of the lemma is finished. Lemma 3 . 5 For -1 < x Â£ 0, i V { i + jJ I t k.a ~ x \ r-2\r 2 C 4 n s J Â» P,(x)P M (t*. a )]| S 2s 2 [P,., ( x)P s U k ' D ) 3-2 L (3.3.12) = O rysinbsinOj^ 2 i + I n 3 sin 3 Â— Â— n for i =3, 4.

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45 Proof s From Lemma 3.4, (3.3.13) Â£ 2s2 [ p *-i p s ( t*. a ) p. I*) P B -i l t k , a ) ] (1 x 2 ) (r + 1) | n / tk.n ~ x \ \P' Z ( X ) P r+1 ( tjj. a ) P r+l (^0 -^x* ( | * (1 ft^ 1 x\Â‘ 11 |P ' U) P -" ( p Â»-i 1x1 p r < tfc J I * jl K?xf 1 Â’ (1 ti Â„) (r + 1) | |2 l P ^ ^ Pr+1 Pr+1 ( ^k.a) P r l x ) \ + r(r + 1) | P r (x) P r+1 ( Â£*.,) P rtl (x) P r ( t k J | + 2 |x t k>1 We now break into two cases . Case 1 : -1 <; t k n <; Â— To get (3.3.12), we need to analyze the order of the following terms . From Lemma 3.2 (3.3.3), (3.3.14) E 1 ( Â— L_ + Ju r =2 ( r 2 n 4 n 5 Â• |p; (x) P r+1 ( t k>n ) pU lx) P r ( t kiD ) | (1 ~ X 2 ) (r + l) Â’ 5 1 It *.*-*! 2 = o = o as r=2 i 1 2 V rn 4 J sin 2 6 (r + 1) (sin0 + sinÂ©*. B ) ' t kin x| 2 ^sin 3 0sin0^ n 3 sin 2 Â— v/sTnBsind^, and

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46 n-i E (1 1 1Â„) (r + 1) (3.3.15) -O o n 5 ) 1 C Jc.n ,) P r+1 (x) PU 1 ( 1 ' | si -n 2 0 fc , n V -^ 4 , 1 1 ^ 1 t k a x\ 2 ^sin 3 Q kn sinQ n 3 sin 2 0 ysinBsinbjj. ^ From Lemma 3.1 (3.3.1), (3.3.16) (1 x 2 ) (r + 1) x ! 3 Â”s + -A) i -2 ^ r 2 n 4 n 5 ) |F r (x) -P f . x _ -Pf+i ( x Â’) -P r ( | = of Â— + Â— ) sin2e < r + D (Sin 8 -IsinB^. n ) ' V n 4 r=2 ( r 2 n) | t k o x | 3 r ysinbsinb*. n t = O n 4 sin 3 AAAii y/smbsinb^ (3.3.17) l) f 1 + jl ) 11 ~ fc *-J (r Â•*Â• x -2 { r 2 n* n 5 ) | t kia x\* \^r ( ^k,o) ~ ^zt-l ( Â£fc,n) -^r (Â•*) | = o = o (AY f 1 + i] s in 2 d ki Â„ (r + 1) (sin0 + sinÂ©*. n ) ^ ^ r2 4 rÂ»2 U 2 n\ 1 1^ k| 3 r y'smOsinb^. ^, J a 4 sin 3 19 ~_ 9 *'Â° ysinbsinb* a and

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47 (3.3.18) n-i 2 = O = o L + 1 ) r (r + 1) 22 4 22 5 J 1 tic,* x\ x) P r+1 ( t k .J P r+1 I*) P z ( t kiQ ) | f n-i 2 f i ' r(r + l) (sin0 + sinS* Q ) r=2 V \rn\ 1 t k.n ~ x\ r ^sinOsinO i a J / 1 ^ 22 3 sm 2 1 ^sinOsint^j Also, (3.3.19) n-i l 2 r=2 r 2 n i (2) = O 22 ' V n 2 ) = o , H 2 V ,sin0sint} i:,n , In case 1, the lemma follows. Case 2 ; Is t* Â„ < 1 2 Jc,fl Since -1 < x ^ 0, we note that |t k(a x| i Â— . An 2 elementary application of estimates (3.2.8) and (3.2.10) along with some calculations on the various terms in (3.3.14) (3.3.19) gives

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48 -i n-i 1 2 C Jc. a ^U=2 1 + 1 V r 2 rz 4 i7 5 (3.3.20) 22s [ P s _ x (x) P x ( P s ( x ) P s _ x ( tjc> J ] s=2 -o A -o n / ^ u 2 y/sinOsinO^. ^ The lemma follows . Lemma 3 . 6 For -1 < x ^ 0, n 4 I tjc,* * ^2 2s 2 [P s _ 1 (x) P s ( t^ n ) P s (x) P s . x ( t*, n ) ] (3.3.21) = O ,/sinO sinO^ 22 2 sin 2 Â— a *'* 1 + 1 22 3 sin 3 e ~ 9 * ,a ^ ^ 2 for i = 2, 3 /

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49 Proof ; We begin by utilizing Lemma 3.4 as in (3.3.13) Y 2s 2 [P s _ x (x) P a ( t k J P a (x) P a . x ( t ka ) ] | P' Q .i (X) P D . itl ( t kia ) (1 x 2 ) (n-i x I tjc.n i + 1) (3.3.22) Pn-i+l (x) Pn-i ( tj :>n ) | \Pn-i Pn-i+l ^ ^k,r) ~ Pn-i+l P n-i ( ^k,n^ \ (1 ~ X 2 ) {a j l^k.n " ^l 2 + ,(1 tl, n ) (n-i+l) . , |t*.Â„ x| \ Pa 2 P Â”-1 I Pn-i+l ( P n (x) r^l + (1 tin) (n-i* 1) ' J 1 K.-xlÂ’ l^n-i ^ ^Jc,.n) Pn-i+l + (n-i) (n Pn-i+l ( tk.n) Pn-i (*> | i) (n i + 1) l^a-i (X) P a .^(t kta ) ' ' v L L ^ | r n-i c n-i+l ~ Pn-i+l Pn-i ^ k.n ) | + 2 |x _ We break into two cases Case 1 ; -1 < t k Â„ <; Â— ^ Â— jc, n /} As in the previous lemma, we need to estimate six different terms. From Lemma 3.2 (3.3.3), _ g _ x 2 ) (n i + 1) | / ( ) , j (3.3.23) ^i+ i (x) ( t* >a ) | 0 n 3 sin 2 Â— " 2 6jc ' al y'sinUsintJ^ and

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50 Â— 1 + 1) [p / ( t )p (x \ Â„ 4 i *. 1 2 r a -2 v c kia ) w ^ 4 | * 12 (3.3.24) Pii+1 (t Jt(n )P fl _ i (x)| = O n 3 sin 2 Â— 2 6 *-^ y/sinOsinb^j From Lemma 3.1 (3.3.1), -li x ^ ^ n j . + ip . (x) p (t) ,4 | tÂ„|3 I n -1 ^n-i+l \ c k.n> n \ C k.n ~ (3.3.25) -^n-i+l (x) p^ ( t k>n ) I = 0 n 4 sin 3 |e ~ z 8jc al y'sinbsinb^ (1 ~ t kin ) (n i + l) 234 1 1 *.* Â“ *l 3 l^a-i (t-k.o) ^n-i+1 (3.3.26) and = o n 4 sin 3 Â— Â— VsmbsinO^ ^

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51 (n i) (n i + 1) ni \t k .n ~ *1 I P n-i (X) P n . i+1 ( t kiD ) (3.3.27) ^a-i+l (x) P D -Â± ( tk, D ) = o 3 ei n2 Â®k,n nÂ° sin ysintfsinO^ ^ Also, (3.3.28) -2<; 1 n 4 k -n 2 ,/sine sinO*^ The lemma in case 1 follows Â£ ase 2 ; j * t k . a < 1 Here we note that \t kfB x\ z and apply (3.2.8) and (3.2.10) to the various terms in (3.3.23) (3.3.28). The result is (3.3.29) n*\t k .n ~ x\ ,S 2 S *{P,_ k ( X) P, ( Â£*.Â„) P, U) P,., ( t k , Â„) ] O n 2 y/sintfsinO^ and the proof of the lemma is complete.

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52 Lemma 3 . 7 For -1 < x Â£ 0, I* Â“ t k.n\ \*k.n U) I = O / sin8 n (3.3.30) n 2 sin 2 ~ 9 *J n 3 sin 3 Â— Â— ^ + 1 n for k = 1, 2, . . . , n-1 Proof ; To begin with, define (3.3.31) a z a = n(n 1) + r(r 1) and k i n 1 a r,n From summation by parts (3.2.1) and (2.2.10), n Â£ (2r (x) i =2 n{n 1) + z(r 1) ?, [^r,n ~ ^r*i,n] ^ (2 S 1) P g _ x (t kn ) p' g _ x ( X ) x -4 8-2 + * n -i.n "s (2 a 1) P a . x ( t k J p' g . x ( X ) (3.3.32) a2 r 2 .= 2 2 r T-2 a ID a r+l,n P U (*> ( tjc. a ) Pr (X) P r _, ( fc fc . a ) C k,n ~ X + 2 *1? ^z-l Pf ( ~ Pf Pf-1 ( 1=2 a r.D a r*l,n [t k Â„ ~ X) 2 _ (n ~ 1) 2 (n l) 2 -1 <*> P n -2 ( + P a 1 <*) ( tfci.) t~. Â— V / j9 Recalling the definition of r k/n (x) given in (2.1.12), we may write (3.3.33) (x t k#n ) r* (n (x) = Z x + Z 2 + Z 3 + Z 4 + Z 5 where

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(3.3.34) (3.3.35) (3.3.36) (3.3.37) and (3.3.38) We begin 4 n(n 1) (l x 2 ) p a-1 (x) (1 tÂ£ fl ) 2 [P^ (t k>a ) ] 3 n-2 2 r=2 a a . r,xi"Â’r+l,xi [Pr-l (*> Pr ( t k J P' r U) ( Â£*.Â„> ] -4n(.n 1) (1 x 2 ) P n-1 (x) (1 t|, fl ) 2 [pi_ 1 (t Jc>n )] 3 c-2 2 1=2 a r,n a r+l,a Pr-^X) P r (t kiB ) ~ P f U) P r _! ( t k . a ) X ~ t k,n (1 X 2 ) P n _ x (x) p' n x ( X ) (1 tin) lPn-l(t ktD ) ] 2 ' ^ = -n(l x 2 ) P . (x) r / 1 (1 t|.jMP'1 (t it . n )]3 U) P 3 Â‘ t *"' ) 1 ' *5 = n(l X 2 ) P^U) r P n _ x (x)P a _ 2 U kia ) (1 tiL ) 2 ^-!^)] 3 fc*. a by working with Z 2 . From (3.2.1), estimates (3.2.7) and (3.2.11), and Lemmas 3.5 and 3.6,

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54 I *2 I = O r 2 8=2 sin-^G^sinG*. n Â‘a3 2 1 1 1 tjc.o r=2 Â®r,a a r+l,n 0 l z*l.n ct z+2,n [P s -i (*> P,U k , n ) Ps (xJP^ft^J] * 1 Â“ n -2.nÂ« % 2s2 [ p s 1 <*> P S < **.!,) P s (X) P,., ( t*, a ) ] ) (3.3.39) = O sin 3/2 8 v /sinB Jc ^ [ tjt.a X\ 's 3 f-L_ + _L) 1-2 ( r 2 .n 4 n 5 ) S 2s 2 [P s _ 1 (x) P s ( t k 'Â„) P a (x) P s . x ( t*, J ] rr o V 2s 2 [P,., (x) P 3 ( t*.*) -P s (x) P,. x ( <:*.Â„) ] sinG n n 2 sin 2 Â— Â— ^ftil n 3 sin 3 n 2 2 We now work with Z x . From (3.2.1) and Lemma 3.3, (3.3.40) z = 4n(n 1) (1 x 2 ) P a-1 (x) ft-ttJ'NiiitfcjJ 1 f S 3 r 2 (r + l) 2 [ r =2 . *r,oÂ“r+l,o Â“m. o*r+ 2,o . J 2 [p^ (X) P, ( Pi (x) P,., ( t fc Â„) } 2 n-2 + ~ 2 ) * 0 2 , 0 * 0 1,11 s=2 = Oi + C> 2 + Â£>3 + Â£ 4 + Q s + Â£> 6 2 [Pi-! (X) P, ( (*,Â„> Pi (X) P,-! ( t t .Â„) ] where

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55 (3.3.41) (3.3.42) (3.3.43) Q _ ~4 n ( n 1) (1 x 2 ) P n _ x (x) (1 t|, a ) 2 [Pa-i (tjt. fl )] 3 % 2 (2 r + l) [n(n 1) r(r + l) ] 1=2 ^r,n Ct r+l,n a r+2,n r(P r . 1 (x) ~ P r (x) ) 1 X *,<**.Â»> + 1 0 2 = 4 n (n 1) (1 x 2 ) P fl-1 (x) (1 J 2 IP.-! ( t t J ] J a j? (2 r + 1) [n(i3 1) r(r + 1) ] a r.aÂ« Â• [Pi (X) P; ( t,.,) ] , Â®r,nÂ®r+l,ziÂ®r+2 ( fl Â£>3 = 4n (n 1) (1 x 2 ) P M (x) (1 O 2 (Â£*,Â„)] n-3 2 r=2 2 (2 r + l) r [j 2 (n 1) r (r + 1 )] a r,n a r+l,n a r+2,n ^k,n ~ Â• [P r . x (X) P r ( P r (X) P r _ 1 ( Â«*.Â„) ] , Â£>4 = 4 n (n l) (1 x 2 ) P n-1 (x) (n 2) 2 ~ 2 [-Pn-1 ( tjc.xi) ] 3 a fl-2,u a j:-l # n 2 )(P fl 3 (x) ~P n _ 2 (x)) 1 X P n-2 ( tk.n^ + 1 (3.3.44)

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(3.3.45) and (3.3.46) -4 n (n 1) (1 x 2 ) P n _ x (x) (n 2) 2 (1 t*. ( t*. a ) ] Â®n2, n a D -l.a [P^_ 2 (x) P n _ 2 ^ ] / -4n (n 1) (1 X 2 ) P fl . ! (x) ( J 2 2 ) 3 (1 t*. Â®n-2,n ' P a _ 3 (x) P n -2 ( tk.n) ^-2 (X) P n _ 3 ( t k .n) t k, a * We now estimate Q x . From (3.2.7) and ( 3 1 0i I = O (sin 3/2 0 ^sinO tffl (3.3.47) n 3 S n 3 JsÂ±nt)slnO k n o sin0 2 . 11 )/ We next skip to Q 3 . From summation by parts (3.2.1),

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57 n 3 2 r= 2 2 2(2r + 1) r [ii (n 1) -r(r + l)1 t *r,.n < *r+l,.nÂ®r+2,.n Â• [P r _ t (x) P r ( t*, a ) P T (x) P r _ t ( t kgn ) ] n-4 2 r=2 (2 r + 1) [n(n 1) r (r + 1)] ^ ^i,n^r+l,n a r*2,n _ (2r + 3) [n(n 1) (r + 1) (r + 2)] Â• -E 2s 2 [P_ t (x) P s ( t fca ) P s (x) P 3 _ t ( tjc> J ] (2n 5) [n(a 1) (n 3) (a 2)] (n 3) a n-3 , d Â® n-2 , D a n-l, n * "I 2S 2 [P S _ 1 (x) P s ( t k> n ) P s (x) P s _, ( t kin ) ] Together, (3.3.43) and (3.3.48) imply that Ii? 3 1 = O n 2 (1 x 2 ) l-P^ (x) | , (1 O 2 | pU ( t kia ) p (3.3.49) n 4 2 r -2 i rÂ‘n A. _ + _A !n * n 5 2 2s 2 [i> s _ 1 (x) P s ( t*. a ) PÂ„ (x) P a . x .rr n-3 2 2s 2 [P s . 1 (x) P s ( t k J P 3 (x) P s _ x ( tjc>n ) ] <**.Â«>]

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58 From (3.2.7), (3.2.11), Lemmas 3.5 and 3.6, along with (3.3.49) , Iftl -o ( sin0 n (3.3.50) n 3 sin 3 Â— Â— n 2 n 2 sin 2 Â— ~ 2 + 1 / Using (2.2.4), we may rewrite Z 3 as (3.3.51) Z 3 = ( n ~ 2) (1 x ) P n _ 2 (t kn ) P n-1 ( x ) P n -i ( x ) (1 t * 2 .=> 2 lH i ( t fco ) ] 3 We now look at the order of the following combination of terms. (3.3.52) Z 2 + z i + Q 5 = Â— Pn ' 2 ^ tk a ' ) Pn ~ 1 ^ (1 tin) 2 [P^! ( tyj ] 3 Â• (n 1) P^j. (x) nPax (x) 4il(Â£1 ~ 1) (n 2) 2 p'. 2 (x) = -d ~ X 2 ) P n . 2 u k , a ) p n x (X) (1 " ti,n) 2 ] 3 ("2n + 3) P 0 '2 (x) {pU lx) + P^2 (x) ) + (n 1) (n 2 3n + 3)

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59 From (3.2.7), (3.2.14), and (3.2.11), (1 x 2 ) | P d . 2 ( t kin ) P a _ x (x) | (3.3.53) (i ti a )*\pUi t k , n ) Pa -1 U) + P'n-2 (X) | = O V n 2 ) From (3.2.8), (3.2.10), and (3.2.11) (3.3.54) ~ y2 ) I F n -2 (tjc.g) Pg-1 (x) I (2i? 3 ) p'n-2. (x) _ q ( sin 6 'j (1 tk.n)*\PL-i (c k ' 0 ) I 3 (n 1 ) (n 2 3 n + 3 ) V n 2 ) Hence, (3.3.55) |Z 3 + Z 4 + Q 5 | = O sin8 n 2 j We next look at another combination of terms + Q* n( 1 x 2 ) P B _ 1 (x) (xc t ,J( 1 cL) ! ipLa^,j 5 3 P n -1 U> P Â»-2 ( t*,Â„> ' 11 l n 2) Â® n-2 , cÂ® n-1 , a (Pa-3 Pn-2 ( P a -2 (x) P n _ 3 ( t kn ) ) (3.3.56) = n(l X 2 ) P D _ X jx) Ut kiD ) (1 tj, a ) 2 [P n / _ 1 (t Jt>a )] 3 Pa1 Pn-2 ^ + -^n-3 -^n-2 ( ^k,n^ P n-2 <*) ^-3 ( tjc n ) + 2U2 + 611 ^ *Â’ a (n l) [n 2 -3 n + 3) (Pa-2 (X) P a _ 2 ( t*. J P n _ 2 (x) P n . 3 ( t*^) )

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60 From (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1), we have (3.3.57) n( 1 x 2 ) | P a _ x (x) | | -2 n 2 + 6 n -5 \ x ~ tk.A (1 tl n ) 2 | PU ( C kia ) | 3 (n 1) (n 2 3 n +3) I P n3 P n-2 ( tk lD ) ~ P n -2 P n-2 ( t kn ) \ Â« o = o ' sin 2 0 v /sTn0^ (sin0 + sinO^) \ x ~ t *. a l n3 sin0^sintt Jt#fl sin0 3 ] e e*. , 22 3 sinÂ— for -l < t v n Â£ Â— k, n 2 From (3.2.7), (3.2.8), and (3.2.11), nil x 2 ) | P n-1 (x) 1 | -2n 2 + 622 -5 | \ x tjc.J (1 tln) 2 \PLl( t k J I 3 (22 1) (22 2 322 +3) (3.3.58) Â• I P a -2 (*) Pn -2 < t k , Q ) P a _ 2 (x) P D _ 3 ( t kiB ) I o sin 3/2 0 v /sin0^) 22 1 = o *Â« i * ^ < 1 From the following identity [32], (3.3.59) (22 2)P n _ 3 (x) = -(22 1) P a _ x (x) + (222 3)xP d . 2 (x) , we may write Pn-2 P n-2 ^ ^zj2 -^c-3 ( Â‘ '' to -' 2 V U Â‘ **.Â«> P -2 Â«*.*> P n -2 r a-l Â’ (3.3.60)

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61 Hence, (3.3.61) From (3.3.62) nil x 2 ) P n _ x (x) (* " C Je,J (! Â“ tin) 2 [P'n 1 (tjfc. a ) ] 3 ^ ^a-l d) -Pq2 d^ >n ) + P n _ 3 (-X) -Pa-2 (t k n ) P n 2 ( x ) P a 3 (t k n ) ] 21(1 X 2 ) P^ (x) Ut*. a ) (1 t*. a ) 2 ] ^ p (x) P ( t ) + d-Q 3 ) (21-2) ^ ^ P n-2^t k>n ) + (J2 _ 2) tjc.a) P fl -2 ( tjc.a) P n 2 U) (3.2.7), (3.2.11), and (3.2.16), 21(1 -X 2 )P n 2 , 1 (x) jP n _ 2 (t^) 1 tjc.J (1 t*,n) 2 |Pi-l(t i(iJ ) | 3 (21 2) Â’= O sin^sinO^ , 21 3 sin0|x t kin \ j o sin0 22 J sm 10-0* /

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62 From (3.2.7) and (3.2.11), (3.3.63) n(l x 2 ) [P^ (x) P n . 2 (t k _ n ) P n _ 2 (x) 1 (2 n 3) (1 tln) 2 \Pn-i(t: k , D ) | 3 (n 2 ) = o ' sin0 nÂ‘ We conclude that (3.3.64) \Z 5 + q 6 \ = Of sin0 1 + 1] \ n < 2 Â• o 1 0 Â” 0 u n 1 n 2 sm 2 hai 2 n / To estimate Q 4/ we use (3.2.7) and (3.2.11) to obtain (3.3.65) |f? 4 | = O sin0 We now observe that we need only estimate Q 2 summation by parts (3.2.1) and (2.2.10), From

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63 ( 3 . 3 . 66 ) JI ' 3 (2r + 1) [n(n l) x(x + 1) ] S r-2 a r , n a rÂ«l , n a r*2 , a P'r (X) P r ( t k _ a ) n(n 1) r (r + 1) _ zt(n l) (r + 1) (r + 2) a r.n a r*l,n a r+2,n a jr*l,D a r.2,n a r+3,n S (2s + l)p'(x) P.(t k J + ~ ^ ~ ~ 3) (n ~ 2) n 4 = s r=2 a zi3 ,/i a n2 ,n a n-l,i 3 n-3 S (2s + 1) p'(x) P^t* n ) &=2 a x 4 (x + 1) [ 2n (n 1) r (x + 2) ] 2 r=2 ^r,nÂ®r>l,n ^z+2,n **r+3,n Â•i + i \ P r (jf) P r . t ( fc^ >n ) -Pr+l -Pf ( ^k.n^ (X + 1) tjc.ii X + (x + 1) Â•Pr+l ^ -^r*l ^ k,n ) _ < t*
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64 = 16 n (n 1) (1 x 2 ) P D _ X (x) (1 tl B ) 2 [Pn-l(t kia ) ] 3 (3.3.69) Z2-4 r=2 (r + l) 2 [2n(n l) r(r + 2)] Â®r,nÂ®r+l,n^r+2,n a r+3,i3 P r (.y) Pj-t! ( -Pfi-i ( t k n ) ( tjt. a X) 2 (3.3.70) -48n ( n 1) (1 x 2 ) P D _ X (x) a O a (*,Â„>] 3 . Â“j 4 ( r + D [2.n(i7 1) r (r + 2)] r=2 a r,JJÂ®r+l,n a r+2,i2 a r+3 / xi R = 8 12 (a 1) (n 2) (2ri 3) (1 x 2 ) P a _ 1 (x) a n-3,n a n-2.n a n-l,n ( ^ ^Jc,a) 2 [-Pn-1 ^k,n^ ] 3 (3.3.71) . ^-3 (X) PÂ„_ 2 ( ~ P'_ 2 (X) P n _ 3 ( t k , n ) ' C k.n ~ * R = 8 n (n 1) (n 2) (2 n 3) (1 x 2 ) P a _ x (x) a n-3.n*n-2.n*n-l.n (1 ~ t|,J 2 [P^ ( ] 3 (3.3.72) . ^n-3 ^X) P n 2 ( ^k.n) ~ ^n-2 ( x ) P n _ 3 ( t kQ ) (tk.n-x ) 2 y and *6 = ~24n (n 1) (2n 3) (1 x 2 ) P a _ x (x) t k a a n-3.n*n-2.n a n-l.n(l " t l. D ) 2 [P'_ 3 ( t*, c ) ] 3 ' (3.3.73)

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65 We proceed to estimate R x R 6 . From (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2), 1^1=0 (jnsÂ±nO kD sin 2 0 (3.3.74) . n jf r2 W sin0 ic, fl + VsinB) ' r=2 n 6 y^\t kin -x | 2 = o / V sin 2 0 (sin0 + sinO^J I t k,n ~ x \ 2 \ / o sin0 3 Â• 2 0 9/c J n 3 sin z 2 for -1 < tk.n * 1 2 ' From (3.2.8) and (3.2.11), (3.3.75) |^|=0 sinO n 2 for s * tfc.n < 1 We move to R 3 . From (3.2.7) and (3.2.11), (3.3.76) 1*3 I We next apply (3.2.8), (3.2.11), and Lemma 3.2 (3.3.4) to R,

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(3.3.77) Also, (3.3.78) From (3.3.79) When (3.3.80) 66 Is. I -O ' sin0 (sin0 + sin0 i>a ) nt k.n x\ = o sin0 n 3 sin Â— for -1 < t k,n 1 2 ' from (3.2.8) and (3.2.11), 1^1 -0(^5) for | ***..<1. (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2), l*s I = o ' sin 2 8 v /sinO fc>a ( v /sinO A . >c + Vsin0) n < t*. a x) \ / = o ' sin 2 0 (sin0 + sin0 Ata ) n 4 I t L ~k,n = o sin0 n 3 sin 2 Â— e ~ 8fc,i: for 1 < t k . a * 1 2 Â’ -r * t k,n < 1/ we use (3.2.28) and (3.2.11) to get l* 5 1 0f-2Â±s5) ( nÂ‘ ;

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67 From (3.2.8) and (3.2.11) , (3.3.81) = O ' sin8 l \ > Finally, we concentrate our efforts on R x . Define (3.3.82) Y _ ~2n 2 (n-1) 2 (2r+3) + 4n(n-l) (r+2) (3r 2 +7r+3) 2r (r+1) (r+2) 2 (r+2) a r, n a r*l , n a r+2 ,n a r* 3 , a a r+4 , n and a (n-3) 2 (n 2 +4.n-8) a a-4 , n a n-3 , n a n-2 , n a n-l , n a-i,n

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68 From (3.3.83) summation by parts (3.2.1) and Lemma 3.3, V ( r + 1 ) 2 [2 n[n 1 ) r(r + 2) ] r=2 Â®r,aÂ®r+l,a^r+2,aÂ®r+3,a Â• [p'U)P rt i(t u ) Pz + 1 Pz ^ 1 a-5 s r=2 (r + l) 2 [2 n(n 1) r(r + 2) Â®r,a ^r+l,a Â®r+2 , a ^r+3,a (r + 2 ) 2 [2a (a 1) (r + 1) (r + 3) Â®r+l,a Â®r+2 ,a Â®r+3 , a Â®r+4,a Â• J 2 [P' (X) P s+1 ( t*, J Pi +1 (X) P s ( t^ a ) ] + (rt ~ 3 ) 2 [2.n (n l) (n 4) (n 2)] a . a , a ^ a . 12-4,23 23-3,22 ^22-2,12 22-1,22 12-4 Â• S [Pi (x) P s+1 ( t*, J Pi +1 (x) P s ( t* >a ) ] S= 2 a-5 S V Â„ I r, a r=2 r(P r . 1 (x) P r (X) ) 1 X *r (**.Â»> .+ 1 p' z (x) P r (t i#a ) . r Pz-1 (X) P z ( t^) p r (X) P r-1 ( fc^) tjt.n " * + O a-4,a (n-4 ) (P a _ 5 (x) ~P a _ 4 (x)) 1 X Pn-i(t k .n) + 1 ^-4 <*> P n -4 (^. n ) Â“ (n-4) Pp-5 Pg-i ^k.n) Pg4 (j^) -^a-5 ( ^-Jc,a^ 'i,a X

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69 We may now write (3.3.84) Ri = S x + S 2 + S 3 + S 4 + S 5 + S 6 where (3.3.85) 16 n(n 1) (1 x 2 ) P n-1 ( x ) (1 ~ Â£k,n) 2 [Pn-l ( tk.n^ ] k.n ~ X ) n-5 S V Â„ * r, n x=2 1 (3.3.86) -I6.n(.n l) (l x 2 ) P n _ x ( x ) (1 P, ( t kiR ) , (3.3.87) S = 16n {n ~ D ~ x 2 ) P n 1 (*) ~ tk.n) 2 [-Pq-1 ( tk.n) ] k.n X ) 2 ' 2 Y r ,n r [ ^r-1 ( tjt.J ~ P r U) P r . x ( t iifl ) ] , (3.3.88) = 16n(n l) (1 x 2 ) P n _ x (x) q n _ 4 >a (1 O 2 ^^)] 3 ^ -x) Â• (fl ~ 4) (P ^ (x) ~ p ^-4 <*> > Â» , +1 1 X *a -4 v Cjc ' n; x = -I6n(n 1) (1 X 2 ) P n _ x (x) o n 4>J d _ tk.n) 2 \Pn-l ( tjc ,n) ] _ x ) [P fl _4 (x) P n _ 4 ( t^ >c ) j , (3.3.89)

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70 and (3.3.90) Sg = ~16n(n 1) (1 x 2 ) P a-1 U) o n _ 4n d Â“ 2 [P fl -i ( t k n ) j ( t ka x) 2 [^n-5 ^n-i ( -^n-4 (x) P a _ 5 ( J . We now estimate the order of the terms S x begin, notice that | Yr .J = O ^ j and |o c _ 4> J = From (3.2.7), (3.2.8), and (3.2.11), (3.3.91) |SJ > o sin 3/2 0^sinO i#JJ tk.a ~ x\ 2 1-2 = o sin0 n 3 sin ~ 9 *-^ \ j Using (3.2.7), (3.2.8), (3.2.10), and (3.2.11), one may obtain (3.3.92) I 3Â»l = O yfc sin 2 e v / S inU J ,
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71 (3.3.93) \S 2 \ O /n sin^sinB^ *-s (^sinO + y/sTnB]Â”) Jz' t k.n x \ 2 1=2 n ' = o sin0 n 3 sin 2 Â° k D for -1 < t k,n 1 2 ' From (3.3.94) (3.2.7) , I 5 3 I = O (3.2.8), and r sin8 ' v n 2 > for 1 2 (3.2.11) , * t k . n < IWe use estimates (3.2.7), (3.2.8), and (3.2.11), to write (3.3.95) IS. | -O sin 2 0 ^ 3 I tk.n ~ x \ \ > -o sin0 n J sm0 k.a 1 From (3.2.7), (3.2.8), (3.2.10), and (3.2.11), (3.3.96) |S 5 | -O sin0 n ' ~k,n ~ X\ -o sin8 n 3 sin 2 * 9 ~ 8 *-Â° 1 \ 2

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72 Using (3.2.8), (3.2.11), and Lemma 3.1 (3.3.2), we have (3.3.97) | S | = q ( sin 2 e^/sinU^ g (y/sinfl + ^sint^ J l n * I t k.n ~ *l 2 o sin0 n 3 sin 2 Â— ~ 9|C,J for 1 < tk.n * 2 m 2 Estimates (3.2.8) and (3.2.11) imply that (3.3.98) l^el =0(^)tor The lemma now follows . Lemma 3 . 8 (3.3.99) \ x ~ tjc.nl kj e.n(*> I = O ( -^^ ) f or 1 0 jc,n ~0| < ,k = 2, 3, ..., n-1 . Proof : We begin by showing that r kn (x) can be written in the form L k.n (X) = (1 X 2 ) pj., (x) (1 t l J (x \ 2 r r-*/ (1 x 2 ) (x) (3.3.100) (1 tk.n) 2 [Pn-l ( t*, n ) ] 3 Â• D Â£ n ( n ~ 1) ~ r(r 1) ' r-z n(n 1) + r (r 1) Â• (2r DP^ftJ^U) .

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73 Notice that from (2.2.4) and (2.2.10), n-1 2 1=2 n(n l) r(r 1) n(n 1) + r(r 1) Â• (2r DP^ItJP^U) = -2 1 (2r l) P z . 1 {t k a ) pU (x) r=2 + 2 n(n 1) 2 (2 r 1) r =2 n(n 1) + r(r 1) * P z-l (tjc.fl) p r-l (*) = (^ 1) (3.3.101) P n _l (x) P n _ 2 ( t^j) + -^n-i (Â•XÂ’) P n -2 ( t kiI1 ) ( tjt.2. 2 tjt.il " * + 2n(n 1) 2 (2r 1) =2 n{n 1) + r(r 1) Pi 1 ( tjj.^) Pi-x ( X ) = (1 ^.JP^Ufc.JPn-iU) tjc .a " * + ^ -Pj-l (Â£k,n^ P n 1 (x) ( t*. a x) 2 + 2n(n 1) 2 (2r 1) r=2 n(n 1) + r(r 1) P x _i ( t k n ) P x -x (x) .

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74 Hence, (3.3.102) r k.n (Â•*) = (1 X 2 ) P n 2 _! (x) kjc,ri) (-X ^k,n) 2 I -Pa-1 ( tjt.a) l 2 (1 x 2 ) P n _ x (x) (1 tJ.a) 2 [-Pa -1 ] 3 (1 ti.a)Pa-l(tjc,a)^a-lU) C *.a * ll r.O Â„ 2n(n . D (<*.Â» *> s a-l 2 Z*2 ( 2 -T 1 ) _ , . . Â—/ . , 22(22 1 ) + r (r 1 ) *~ x {C k.ni*z-iM m (1 x 2 ) P n-1 (x) p' n . x (x) (1 tÂ£j (X t kia ) [Pn-l(t kiD )] 2 _ 2n(n 1) (1 x 2 ) P a-1 (x) (1 tk.n) 2 lPU(t k . a )l* . "f 1 <2r l)P rt ( t kin ) PU (x) r-2 22(22 l) + r(r l) From (2.1.12), we see (3.3.100) is correct. Next, we calculate the order of |x t k Q \ \r k n (x) |. Recalling (3.2.2) (3.2.5), (3.2.7), and (3.2.19), we state

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75 (3.3.103) From (3.3.104) From (1 x 2 ) (x) d ~ tk.n) \ x ~ tk.n I ^k,n) ] 2 = o ' (l X 2 ) p n . x U) ' k ^ Â” tk.a) P n1 ( tk.n) t -o sin8 y/sinO a V sin0 *.a ; .2.2), (3.2.3), and 1 0 0^ n | < we have sin0 _ sin0 sinOj. a sin0 *,* sinQ k.n = 1 + sin0 *, n s 1 + 2 sin 9 " 9 *-Â°^ sin0 k.n = O (i) . .3.103) and (3.3.104), (i ~ x 2 ) fj.! (x) _ q/ sin8 (i tÂ£ a ) |x t*J CP^-i (t^) ] 2 la (3.3.105)

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76 Recalling that |6 e^J < , we have \x t k.a 1 9 + 0. Â„ . 9 0 t I = Sin sm-i hÂ£i 2 2 (3.3.106) s [sln6cosJ^^l . | COS0 | o ' sin0 n nÂ‘ It follows from (3.2.7), (3.2.9), and (3.2.11) that (1 x 2 ) | (x) (3.3.107) (i tl n ) 2 \pUu k , n ) |D-l 2 1=2 n[n 1) r (r l) n{n 1) + r(r 1) Â• (2r 1 ) | P x X ( tjc.J P'x-i (x) | = 0(1) Using (3.2.7), (3.2.8), (3.2.10), and (3.2.11), we obtain an alternative estimate . (1 x z ) | P a-1 (x) | (! Â“ tk.n) 2 I P n-1 ( tjc.n) P (3.3.108) i) r (r 1) ' r =2 n (n l) + r (r 1) Â• (2r 1) | P x _ x (t kn ) p/_i (x) | = O (nsinG) . From (3.3.105) (3.3.108), the lemma follows.

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77 Lemma 3 . 9 (3.3.109) | (1 x) r n a (x) \ = Q | S1 ^ n9 j for -1 < x < l. Proof : Using (3.2.7), (3. 3. HO) t 1 ~* 8> | = Oj-SiSÂ®)Define k{x) = (l x) (1 x 2 ) P fl _ 1 (x) % 2 (2r 1) P r y _i (x) (3.3.111) r=2 1) + r(r 1) + ^ 2 (1 X> (1 * 2 > Pn-1 <*> PU (X) . Recalling (2.1.13), we see that it is left to show (3.3.112) |JeU) | = O As in Lemma 3.7, define (3.3.113) a ID = n(n 1) + r(r 1) .

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78 We next apply (3.2.1) to the following S 2 (2r 1)PU (x) = -3 r =2 n(n 1) + r(z 1) r =2 a r.n Â® r+l.n S (2s 1) pj_! (x) + 3-2 n{n 1) + (n 2) (n 3) (3.3.114) Â• 2 2 (2s 1) p' s x (x) = 2? s=2 x=2 Â®r,nÂ®r+l,n S (2s 1) p'_! (x) P s _ x (1) + s=2 2 (n 2 3n + 3) n-2 2 (2s 1) Pg_! (x) P s-1 (1) . s=2 From (2.2.8) and (2.2.10), S (2s 1) pj_! (x) P s _ x (1) s=2 (3.3.115) ' = r = i pU U) pj (x) + P r -! U) P r (x) 1 X (1 x) 2 Pr1 U) ~ Pr (X) PU (X) + p' (X) 1 X 1 X Pr-lU) P'AX) 2 P'(x) (1 X) + (1 X) ' Hence, (3.3. 116 ) ic (x) = Aj_ + ^ + A 3 + A a + A s where

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79 (3.3.117) A x = (1 " X 2 ) P n _ x (x) * 2 3 2r(r + 1} [P'.i (x) p' (x) ] , (3.3.118) A, = (1 X 2 ) P n . x (x) Â• S 4r H . r ~ 2 a r,n a r+ l,n (3.3.119) A 3 (i3 1} (1 X 2 ) 2 (n 2 3n + 3) Â• P^ (x) [p ' n . 3 (x) P^_ 2 (x) ] , (3.3.120) A Â‘ = (n 2 3n + 3) (1 Â• x2) P -Â‘ {x) P Â»2 (x> ' and (3.3.121) ^ = 2(Â” -l) 2 (1 ' X) (1 Â‘ X2) P Â»-> (X) P 2 (X) From (3.2.8) and (3.2.10), (3.3.122) |A,| =o[sin 2 eÂ°2 /! J-0( sia8 ) r=2 n 4 sm0 j \ n 1 and (3.3.123) 14.1 =0(-5i^)

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80 Using (3.2.1) and the observation that we have a telescoping series. 2 3 -r(-r + 1) r Â“ 2 Â®r,aÂ®r+ l,n [Pr-iW -P'rM] D4 2 r=2 r(r + 1) _ (r + l) (r + 2) Â®r+l,flÂ®r+2,xi (3.3.124) 2 [Pi-iU) S-2 p' (x) ] + ll] 3 > 2 > a a * n-2 , n-2 , n n 3 2 [Pj.i (x) p' (x) ] s=2 = _ n 2 4 2 (r + 1) [n(n 1) r(r + 2) ] 1= 2 Â®z,z:Â®r+l, uÂ®x+2,ij [1 P' (x) ] + 2)_ [;L _ p^_ 3 (x) j ^n-3 ,n^a-2,n Together (3.2.8) and (3.2.10) along with (3.3.117) and (3.3.124) imply (3.3.125) K| = O n 4 sin 2 0 2 sin0 ' r =2 n 3 sin0 n 0(^2) Lastly, A 3 + ( n l) (1 x 2 ) P D . X (x) (3.3.126) 2 {n 2 3n + 3) * [i ^-3 (x) Pa-z (x) + (1 X) Pa -2 (x) ] (~2n 2 + 6n 5) 2 (n Â— 1 ) 2 (n 2 3ii + 3) (1 X 2 ) (1 x) P D . X (x) p' n . 2 (x)

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The estimates (3.2.8) and (3.2.10) imply that 81 (3.3.127) Â— Â— Â— + (i x 2 ) (1 x) P 2 (n l) 2 (n 2 3n + 3) ( } ^ (x) n-2 ( X ) 0 (^ 5 ) Using (2.2.5), we obtain P'n3 U) ~ PÂ‘n -2 (X) + (1 X) P' n _ 2 (x) (3.3.128) = P ' n 3 (x) xp' d . 2 (x) = (n 2) P c _ 2 (x) . Hence, from (3.3.128) and estimate (3.2.7), (3.3.129) 3 ) (1 ' x! > P Â« < x > sin0^ * (1 x)Pi 2 U)] =0(^52) We conclude that / (3.3.130) . a 5 | = 0(-2i25) , and the proof is completed. Lemma 3.10 (3.3.131) n s *=0 X tjc.J |r* (n (x) i = O for -1 Â£ X Â£ 1

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82 Proof : We begin by noting that we need only prove the result for -1 < x < 1 since (1 + x)r on (x) and (1 x)r n>n (x) vanish at x = Â± 1 , and r k , n (Â±1) = 0 for k = 1, 2, . .., n 1. We now look at the terms |l + x\ \ r on (x) | and 1 1 x | | r nin ( x ) j . From Lemma 3.9, (3.3.132) |l x\ | r nD (x) | = Q for -1 < x < 1 . Since r D/n (-x) = r 0/n (x) from (2.1.13), we have (3.3.133) I 1 + X l l r o,o< x > I = I 1 (-*) I !*Â„,,,(-*> I = O Â« for -1 < x < 1 . Hence, it is left to show that (3.3.134) Â”s \x t fc J \r kia Or) | = O (-SM) for -1 < x < 1 . Actually, we shall see that we need only prove the above inequality for -1 ^ x < 0. It follows from symmetry and uniqueness of the r x,n(x)'s, that (3.3. 135 ) r kn (-x) =i n _ k n ( x ) and t kn = -t n _ kn for k = l, 2, .... n-1 .

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83 Further , n-1 ^ | (~x) t kia \ \r kin (-x) n-1 (3.3.136) =S \x t a . ktB \ \r B _ ktB (x) k=l n-1 = 2 \x t kiD \ \r k (x) k=l Therefore, proving (3.3.134) for -1 < x Â£ 0 is sufficient. To begin. 2 I* tjc.J I Jc=l (3.3.137) 2 \ x tjc.J I i e 0 *. n i < n + 2 \ X fc*.J l r Jc, D I Â• i 0 0 *. D i = i From (3.2.6), there are a finite number of different 0k, n ' s such that |0 e^J < -Â£ . From Lemma 3.8, we may conclude that 2 l e Â“ e *,al < (3.3.138)

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84 From Lemma 3.7, S 1 8 -a*., I _C n I* t k .n\ I *k.n U) I (3.3.139) = O (le-e^l * | sin6 n n 2 sin 2\Â±^A n 3 3 in 3 9 ~^-' 1 + 1 n After applying (3.2.17) and (3.2.18) to (3.3.139), we have (3.3.140) \x t lÂ©-0jc. n l * n k,n I k,a (x) I . O (^5) The proof of the lemma is now complete. Lemma 3.11 For -1 s x s 1, Â» (3.3.141) S |r fca U) | = 0 ( 1 ) Â• Jc=0 Proof: From the definition of r k(I1 (x) in (2.1.13) and the estimates (3.2.7), (3.2.8), (3.2.9), and (3.2.11), we have (3.3.142) \r na (x) \ = \z on (-x) | = Q (1) for -1 < x < 1 .

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85 Note that because 2 |r^ n (Â±l) | = 1 , it is sufficient to Jc=o ' prove (3.3.141) on the open interval (-1,1). Following an argument similar to (3.3.136), we have (3.3.143) S 1 I | -SÂ’ |r t .Â„(x) |. k = 1 k=l D-l We are, therefore, left with the task of showing (3.3. 144) ^ \r k a (x) \ = Q (l) for -1 < x <: 0 . From Lemma 3.8 (3.3.100), we may write (3.3.145) r k/n (x) = D : k,n where 'k,n (1 ~ X 2 ) P*_ x U) d tk.n) (-X ~ t kiB ) 2 [P a _]_ ( tk, n ) J and J k,n (1 x 2 ) P a _ k (X) (1 tln) 2 \.Pn-l(t kin )] . D jf \ n {n 1) r (r 1) x =2 n (n Â— 1) + i (r 1) Â• (2 r 1) P r . x ( t k n ) P^-i (x)

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86 A result of Turcin [11] states that n Â£ (1 x 2 ) PlX (x) 145 ) 1C=1 d ~ tk.n) ( x tk.n^ 2 t-^n-1 djt.n) 1 2 = 1 Pn 1 U) =0(1). Hence, we need only show that (3.3.147) 2 \D k \ = O (1) for -1 < x <; 0, Jc=l and the lemma is proved. We now restrict x to the interval [1 / 1 ]. From (3.3.107), (3.3.148) K.J = O (1) . We proceed to obtain an alternative estimate of D k>n . Recall that we earlier defined (3.3.149) ,a r n = n{n 1) + r(r 1) . From summation by parts (3.2.1) and (2.2.10), we obtain (3.3.150) y k,n (1 x 2 ) P a . t (x) (1 tin) 1 [Pn ] : s | n(n 1) r(r 1) _ n(n 1) r(r -t1) a [ n(n 1) + x(x 1) zi(n 1) + x(x 1) Â•{:Â€[ Â• i (2s 1) ( t k n ) pU (X) + [ n [ n x) (n -l)(n2) *=2 [ n(n l) + (n 1) (n 2) ^ Pe-i (t k ,n) Pa 1 (*) | = ^i,fc + ^2,Jc + ^3,Jc + ^4,*

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87 where 422 (22 l) (l x 2 ) p n ^ ( x ) a t/L> 2 tiÂ£i< x> r^r Â— [*Â« <Â•*) ^ < <*. Â„> P * ( t *.Â„> ?, <*> ( tfc.) ] . r Â“ 2 tt r,nÂ“r+l,j 2 J >3.k -(1 x 2 ) P a _ x (x) (1 tin) 2 iPLl ( t*. a ) ] P n _l (x) P n ~2 ( tfe-fl) 4, Jc -(1 x 2 ) P fl-1 (x) (1 t*.J 2 [P^-i (Â«*.Â„> ] 3 -^n-1 Pg-2 ( tk,n) ltk.n ~ *> 2

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88 From (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1), (3.3.151) ISwtl o = O sin 6 a j? r (sin0 + sin0^ a ) ' V I ,n *! 2 r=2 n' n 2 sin 2 Â— Â— for "I < t k . n Â£ 4 2 / We use estimates (3.2.7), (3.2.8), and (3.2.11) to get (3 3.152) K.|-0(i)fori,t fc .
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(3.3.155) 4 n(n l) (1 x 2 ) P D _ X (x) (1 tl n )*[p' a _ At k ' D )V(t kiQ x) J fÂ„3 1 (N (r + l) 2 1 , r = 2 Â®r+l,n^r+2,n . Â• j 2 [P'. x (x) P s ( t k>n ) P' s (x) P s _ t ( ] + ~g~ ~ 2)2 ^ U) ( c *. a ) ^ (*> P.-i ( t kiQ ) a n-2,n a n-l,n Â«=2 4n(.n ~ 1) (1 ~ X 2 ) P n _ x (x) (1 -X) n j 3 ( 2r + 1) ~ 1) ~ r(r + 1)} r=2 Â®r,n Â®r+l,n Â®r+2,n r> Pr < J 1 Pi U> P, < tic..) _ r Pr-l U) P, ( t t ,Â„) P, (X) ( t t .Â„) fc Jc,* * + (n 2 ) 2 ^n-2,n^n-l. (n 2) (P Q _ 3 (x) P n _ 2 (x) ) 1 X Pn 2 ( tfcj 1 P n -2 Pn-2 ( tk.n) 2 ) ^~n-3 (x) Pg-2 ( Pq-2 (x) Pn-2 ( ^ fl) t k.n ~ X

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90 We use estimates (3.2.7) and (3.2.11) to get 4n in 1) (l x 2 ) P n _ 1 (x) (3.3.156) tk.n) 2 [-Pa-1 ( ] 3 ( t-k,n n ' 3 (2 r + 1) [n (n 1) r(r + 1) ] 2 r-2 a r , a a t+ 1, n a r+2 , n Pr(tk.n) + 1 r(P r . 1 (x) P r (x) ) 1 x + (a 2 ) 2 Â®a-2,nÂ®n-l,a (T1 2) (P a _ 3 (x) P n _ 2 (X) ) = o 1 X ' sin6 > , n2 \ tk.n * *1, Pa 2 ( ^k,a^ + 1 o 2 Â« o | Q Â— Q _ | n 2 sin 2 2 From (3.2.7), (3.2.9), and (3.2.11), 4 a (a 1) (1 x 2 ) P n _ x (x) tk.n > 2 tPfl-l J 3 ( tjc a-3 2 r= 2 2 (2r + 1) [n(n 1) r(r + 1) ] (3.3.157) Â®r,a Â®r+l,a Â®r+2,n p' (x) P r ( t k n ) + Â— 2)2 f a -Â±^.. Pg ~ 2 ( ^n) o Â®a-2,.nÂ®a-l,fl f 1 1-0 f 1 \ l ^ 2 I t*.a *l 2 J sin 2 A 8 ~ 8 *al k 2 ;

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91 We next use (3.2.7), (3.2.11), and Lemma 3.1 (3.3.1) to get 4 n(n 1) (1 x 2 ) P g . 1 (x) (1 O 2 iP'n-i ( t k J ] 3 ( t kiD x) 2 n j? (2 r + 1) r [rdn 1) r(r + 1) ] . r ~ 2 Â®r,n a x.l,jj a r.2,a (3.3.158) Â• [P z . x (x) P r ( t kiD ) P r (x) P r _, ( t* >n ) ] + a" ' Â” "tt 2)3 U) (t *.J a n-2,n a n-l,.n ^2 U)p fl . 3 (t*. a )] J sin0(sin0 + sinQ*. Â„) = o = o n2 I c jc. 1 a. for -1 < t v Â„ s Â— Jc, n o From (3.2.7) and (3.2.11), 4n (n Â— 1) ( l Â— x 2 ) P a-1 (x) (1 tln) 2 lPn-lU kin )V (t k 'Â„ X) 2 (2r + 1) r [n(n 1) r(r + 1) ] . r Â“ 2 a r,a a r+l,n a r+2,xi (3.3.159) Â• [Fr-i (X) P T ( t t>D ) P r (X) P r _, ( ] * (n 2) 3 [P n 3 (X) P fl _ 2 ( tj. n ) Â•Pfl-2 Pfi-3 = 0 (i ) f01 i 5 !Â•

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It follows from (3.3.155) (3.3.159) that 92 (3.3.160) \S lik \=Q n n 2 sin 2 ~ e * , J f Â° r I 0 0 k,n I * 2 Hence , (3.3.161) I^.J-0 1 + n n 2 s in. 2 Â— 2 We now break into two different sums as follows, (3.3.162) 2 I^J = 2 | D kin \ + 2 |^J From (3.2.6), the first sum on the right hand side of (3.3.162) contains a finite number of terms. Hence, (3.3.147) implies that (3.3.163) 2 | D kn \ = O (1) . From (3.3.161) and (3.2.17), (3.3.164) 2 1^1= 2 ' |8-8fc=l>| The proof of the lemma is now complete. n 10 " = O (1) bio

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93 Proof of Convergence Results We begin this section with a result of DeVore, Theorem 2.4 in [10]. Let Lip 1 denote the class of Lipschitz one functions and C[-l,l] the class of continuous functions on the interval [-1,1]. Theorem 3 . A Suppose that L n is a bounded linear operator on C[1,1]. If Cj i 1 is such that for each g e Lip 1, we have (3.4.1) | L D (g;x) g(x) | * C x ^' 1 ~ x for -1 Â£ x si . Then for each f e C[-l,l], we have (3.4.2) | L n (f;x) f(x) Â± C 2 to for -1 sx si, where C 2 is independent of f, x, and n. The preceding is a specific case of the more general theorem stated by DeVore. We next make the following observation which follows from the uniqueness of the r kn (x)'s. p r k .n s 1 (3.4.3)

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94 Proof of Theorem 3 . 1 Let f Â€ Lip 1. Then using (3.4.3) we have | R n (f;x) f(x) | = p r k . a (x) fix) k=Q (3.4.4) 2 q [f(t kia ) fix)] r kn (x) * p X |x t k-a \ \r k ix) k=0 for some absolute constant k. Hence, using Lemma 3.10, |i? n (f;x) fix) |=OS | x Â— t kin \ \r kin ix) (3.4.5) = 0 jÂ£Z X* n Jc= o for -1 ^ x ^ 1 . From Lemma 3.11, the operator Rn is bounded. Hence, the theorem follows from Theorem 3. A.

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CHAPTER FOUR LACUNARY INTERPOLATION BY SPLINES Introduction and Main Results In the last two chapters, we considered Birkhoff interpolation problems using polynomials. The object of this chapter is to consider analogous problems using spline interpolation instead of polynomial interpolation. We shall limit ourselves to problems in the (0,1,3) and (0,1, 2, 4) cases. Before presenting our main results, we introduce some notation. Denote by S^ ] q the class of splines S(x) such that 1) S ( x) Â€ C r [a, b] 2) S(x) is a polynomial of degree q in [x Â± , x 1+1 ] ; i 0, 1, ..., n * 1. Throughout this chapter, 0 = x 0 < x 1 < ... < x^ < x n = 1 will represent the joints of a spline. Also Â— + 1 Â— 2z i Â— / and 5 ~ max (h^) . Â• i = 0, 1, Â— , n-l The following are our main results. Theorem 4 . 1 Given arbitrary numbers f (j) (x Â± ), i = 0, 1, ..., n; j = 0, 1, 2; f (iv) (z i ), i = 0, 1, . .., n 1 where 95

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96 2 *i = *i + f"' ( x 0 ) , f'" ( x n ) there exists a unique S a (x) e such that (4.1.1) Sa J) {xj = f (j) (x s ) , i = 0,1, . . . , n; j = 0, 1, 2 Â• S^ iv) ( z i ) = Â£ u* ( Zi ) , i = 0,1 n 1 . S"' (X Q ) = f'"(x Q ) , SfUJ =f /// (X fl ) Theorem 4 . 2 Let f e C 1 [0,1] , I Â£ 4. Then, for the unique spline S n (x) associated with f and satisfying (4.1.1) we have (4.1.2) |s n (r) U) fW (x) | s c r#J 8 J r to (Â£<-", 6) , r = 0,1,2, 3,4, 1 = 4, 5, 6, 7, 8. Also, for f e C 9 [0,1] |sj r> (x) f<*> (X) I r = 0, 1, 2, 3, 4 . Theorem 4 . 3 Given arbitrary numbers ffzj for i = 0, 1, . .., n 1; f (3) (xi) for i = 0, 1, ..., n where j = 1, 2, 4; and f(x 0 ), f(x n ); there exists a unique spline T n (x) e such that

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97 (4.1.4) T n (z i ) = f (z j) ; i = 0,l, . . . , n 1 Â‘ T^ J) = fWI (x i ) ; i = 0 , 1 , . . . , n; j = 1 , 2 , 4 r a (x 0 ) =fU 0 ), r a (x a ) = f(x a ) Theorem 4 . 4 Let f e C 1 [0,1], Then for the unique 8 th degree spline T n (x) associated with f and satisfying (1.4), we have (4.1.5) |r a (r) (x) f (I) U) | <; a r l b 1 '* co (Â£Â«>, 6) ; r = 0, 1, 2, 3, 4; 1=4, 5, 6 , 7, 8 Also, for f e C 9 [0,1], (4.1.6) | r n (r) U) fW U)| s a r 9 6 9_r Q Â“ 1 |fÂ»> U)|; r = 0, 1, 2, 3, 4 Next, we present our results for the (0,1,3) case. Theorem 4 . 5 Given arbitrary numbers f (j) (Xi) for i = 0, 1, ..., n where j = 0.,3; for i = 0,1, ..., n 1 where j = 0 / 1 ; and f (x 0 ), Â£' (x n ); there exists a unique spline R n ( x ) e such that RÂ± j) (x,) = fO) ( X .) , i = o, i n; j = 0,3 (4.1.7) (z.) = fO) ( Z . ); i = o, i n i ; j = o,i i?'(x 0 ) = jf'Uo), i?'(x a ) = f'U a )

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98 Theorem 4 . 6 Let f e C 1 [0,1]. Then, for the unique 7th degree spline associated with f and satisfying (4.1.7), we have; (4.1.8) | R< r) ( x ) (x) | s P r/i 6 2 r (o (f< 2 >, 5) r = 0, 1, 2, 3; 1 = 3, 4, 5, 6, 7 Also, for f e C 8 [0,1], (4.1.9) |i? n (r) (x) fW U) | <; p r>8 5 8_r Q ^ 1 |f (x) |; r = 0, 1, 2, 3 Before continuing with the proofs of Theorems 4.1 4.6, we make some remarks. The splines discussed in Theorems 4.1 4.6 involve the use of midpoint data. It would be natural to ask why not interpolate data only at the joints. The preceding question is referred to in a remark by Demko [9] that, as in (0,2) interpolation by splines, the (0,1, 2, 4) and (0,1,3) cases are ill-poised for splines that interpolatedata only at the joints. By allowing the additional freedom of prescribing some data at the midpoints of the joints, we are able to obtain interpolating splines with the following positive properties . 1) The splines given in Theorems 4.1, 4.3, and 4.5 exist uniquely for any choice of knots. 2) As shown in the proofs of Theorems 4.1, 4.3, and 4.5, our splines are uniquely determined by a tridiagonal dominant system. In Theorem 4.1, the

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99 spline is actually determined by a diagonal system. This spline, consequently, is a locally determined process. 3) Unlike many G-splines and piecewise polynomial schemes, our splines are "very" smooth. The (0,1, 2, 4) splines are of the class C 4 [0,1], while the (0,1,3) spline is of the class C 3 [0,1]. 4) Theorems 4.2, 4.4, and 4.6 show that our splines converge with the optimal order of convergence for splines of their respective classes. Proofs of Theorems In this section, we present proofs of the Theorems 4.1 4.6. Proof of Theorem 4.1 If Q(z) is any polynomial of degree eight on [0, 1], then we have Q(z) = '(0)B 0 (z) (4.2.1) C'(l)B 0 (l-z) +Q"(Q)C 0 (z) + Q n (1) C 0 (1 z) + Q , "(0)D 0 (z) Â£>'"(1) D 0 (1-z) + Q (iv) (1) E 1 (z), where A 0 (z) = 1 35^ 4 + 84z 5 70 z 6 + 20z 7 , B 0 (z) 56 z 6 + 3

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100 Co (2) -Do ( z ) and E x (z) + Next, we let x = x i + tJi i# h Â± = x i+1 x Â± , 0 s t z 1 . our object is to prove that there exists a unique S D (x) e satisfying the conditions of Theorem 4.1. define S n (x) by the following representation. s n U) = fiA a (t) + f i+1 A 0 (1t) + h i f 1 / B 0 (t) (1-t) (4.2.2) +Aifi / C 0 (t) + hjf^C Q (1-t) + hlsf? { Xj ) D 0 (t) hi s"' (X i+1 ) D 0 ( 1-t) + h 4 ^ ( t) 2 where fi P) = f ip) (X Â± ) , Â£g{ = fW u i+1)/ = fM (z.) 2 Next, we set Here, We (4.2.3) s'"( 0) = f"'(0), Sf(l) = f"'(i).

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101 Clearly, S n (x) as defined by (4.2.2) belongs to C 3 [0,1], no matter how we choose s'" (x i ) , i=l, 2, ..., n 1 . They are uniquely determined by the conditions (4.2.4) S a (iv) (x d +) = S n (iv) (x i -) , i = 1, 2, ..., n 1. A simple computation shows that (4.2.4) gives (4.2.5) 5h?h?_i ( h i + h^) s'" U,) = 210 (-hi-i -f i + 1 + (hi " *i-i) fi) 7 0 hih^ (hl x f( +1 + hlfi.! + 2 (hi + hi 3 .,) fi) + Shjhl^ hjf'U + 8 (hi hi.!) if) + | hi hi.! Â“2 f . ( i2j Thus, S n (x) defined by (4.2.2), (4.2.3) and (4.2.5) satisfy all the conditions of the theorem. This completes the proof of Theorem 4.1. Proof of Theorem 4.2 To begin, the following identities are valid: (4.2.6) A 0 (t) + A, (1 ~ t) = 1, A (1 t) + B 0 (t) B 0 (1 t) = C, A (1 t) 2 B 0 (1 t) + 2C 0 (t) + C 0 (1 t) = t 2 , A, (1 t) 3B 0 (1 t) + 6C 0 (1 t) + 6 D 0 ( t) 6Â£> 0 (1 t) = t 3 A (1 t) 4B 0 (1 t) + 12 C 0 (1 t) 24Z5 0 (1 t) + 24B X ( t) = t 4 A (1 t) 5 B 0 (1 t) + 20 C 0 (1 t) 60 D 0 (1 t) + 6 0 A ( C) = t 5 A (1 ~ t) ~ 6B 0 (1 t) + 3 0 C 0 (1 t) 120B 0 (1 t) + 90B 1 ( t) = t 6 A (1 " t) 7B 0 (1 t) + 42 C 0 (1 t) 2 10 ,D 0 (1 t) + 105 A ( t) = t 7 A (1 " t) 8 A (1 C) + 56 C 0 (1 t) 336 D 0 (1 t) + 105 A ( t) = t 8

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102 The identities can be derived from (4.2.4) and the uniqueness of this interpolation formula. Next, from Taylor's formula we have ,a -> ,, f m,. c , -V f u> h i-i . f,p 1 (4.2.7) t U lrl ) jS A, j^In , (4.2.8) + ^ (p) (T) 2>J ) (p-i) ! (4.2.9) * 7=4 (J-4) ! V 2 , f (p) hAP-i (p-4) ! and (4.2.10) f 5 '^ (z-1) = S 7=4 = 1 f (7) ( Xi ) (-%f (j-4) ! hi.AP-4 + ( 14> (-^) (p-4) ! where x Â± < Ti lrl < x*^, x^ < r| 2/1 < x Â± , x* < t\ 3 < z lf and z i-l < t) 4 < x 4 .

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103 Let t = X Xl . Then h i f(xl f , + (x-x,) f' * U ~^ ) 2 f" + U ]* J> 3 ff (4.2.11) + * f<Â« <^ 0 ) = f. + U2.fi * + ( t-hj ) 3 fin + ( thj) 4 3! f Â• Let feC p [0,l], 4 Â£ p ^ 8, then from (4.2.5) and (4.2.7) (4.2.10), \f'" (Xj) -s'"( Xi )l * C JL h i\i + h ?-*) a)(f(p), 6) and h i + i for f e C 9 [0,1] \f"'(x i) sf U,)| (4.2.12) c p h i h i _ 1 {h i + Aj-i) max , (9) . + At-! O^x^l l r (x) I ' where 5 max ( h i) and a (f,5) denotes the modulus of i0 , 1 22-1 continuity of f. Next, we let x = x Â± + th Â± , h Â± = x i+1 x i and f e C 4 [0,1] Then using (4.2.2), we obtain (4.2.13) s a (x) f(x) = ^i(t) +n i (t) where

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(4.2 and (4.2 (4.2 Also (4.2 104 .14) 1,(0 =hUSn(x i ) -f'"( Xi )) D 0 ( t) hi (sf U i+1 ) f" U i+1 )) X? 0 (1 t) Â»*i (t) = fiA) (t) + f i+1 A, (i t) + h, (/iB 0 (t) , 15 ) f'i* iBo (1 t) ) + hj [f'i'Cg (t) + filiCo (l t) ) + hi (-ffn 0 ( t) (1 t) ) + hif E 1 (t) Â£ (x) . 2 Next, on using (4.2.12) and (4.2.14) we obtain .16) 11,(0 1 s qi!, 4 0 ^ 1 |D 0 (t)| u(f (4) , 5) . , from (4.2.6), (4.2.7), (4.2.11), and (4.2.15), we have Hi (t) = f t A 0 (t) + A 0 (1 t) (t) h 1 i IjUjI . f Uvl Â„} r l' ! 4 I 1 W-0 J 3 ^Â’nr 1 . f Â«Â•Â» (TU.J Â„3 iÂ£ Ul) l 3 I (1 t) 17) Â• + hi f'lCg ( t) + hi f'l'Dg ( t) + hi f (i ? E 1 ( t) 2 + hlc 0 (1 t) fl U) hi i-2 ^ -f (iv) (n ll2 ) 2 ! J 2 (j 2) ! hi [if + h,f (i * (t| 1<3 )] D 0 (1 t) = hi '"jV BÂ„ (1 t) + _ ^ U * (Tli.a) Cq (1 t) fÂ«*l (t, 1<3 ) D 0 (1 fc) + f
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105 On using the identity given in (4.2.6) corresponding to t* , we have Hi ( t) = h* f (iv) (tii.p) f (iv) (n 0 ) 24 (1 t) (4.2.18) ' f Uv) (Tl 0 ) f (iy) ( T| , ( fUV) (n i>2 ) f UV) (Tl 0 ) S 0 (1 t) c n (1 t) + ( f Uv) (T 1 0 ) f Uv) (n lf3 )) d 0 (1 t) + (^ (iv) iZi) f Uv) (Tl 0 ) ) ^ ( t) ] . From this we obtain (4.2.19) 1^ (t) | s C 6 hf 0) (f (iv) , 5) On combining (4.2.13), (4.2.16), and (4.2.19), we obtain (4.1.2) for r = 0, 1=4. Proof in the other cases are similar. We omit the details. Proof of Theorem 4 . 3 We begin by making the following observations; If Q(z) is any polynomial of degree eight on [0,1], then we have G(t) = 0(0) A 0 (t) + 0(1) A 0 (l t) + 0(-|) A x (t) (4.2.20) O'(0)B 0 (t) Q'(l) B 0 (1 t) +0 /, (O)C o (t) Q"(l) C 0 (1 t) + Q (i) (0) i? 0 (t) + Â£>Â«> (1) E 0 (1 t)

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106 where 1A 0 ( t ) = -384 t 8 + 157 8 t 7 2323t 6 + 1299 t 5 177 t 3 + 7 ( t) = 76 8 t 8 301 2 t 1 + 4352 t 6 2304 t 5 + 256 t 3 7 B 0 ( t) = 122 t 8 + 509 t 7 766 t 6 + 443 t 5 7 1 1 3 + 7 t 14 C 0 (t) = -26 t 8 + lilt 7 17 3 t 6 + 106 t 5 25 t 3 + 7 t 840Â£q (t) = 10t 8 Alt 1 + 87 t 6 79 t 5 + 35 t 4 6 t 3 Let x e [x Â± , x i+1 ] . Then x = x Â± + th Â± where 0 Â£ t Â£ 1. Our goal is to prove that there exists a unique spline T a (x) e a satisfying (4.1.4). We define T n (x) f for Xi * X s x 1+1/ by T n (x) = T n (x^ A 0 (t) + T Â„ (x i+1 ) Aq( 1 t) + f (z Â± ) \ ( t) + B 0 ( t) (x i+1 ) B 0 (1 t) + Alf" (x^ C 0 (t) (4.2.21) + hlf"( Xi ^) C Q (1 t) + hjf < 4 > (x^ B 0 (t) .+ h 4 f (4) (x i+1 ) B 0 (1 t) Next, we set T n (x 0 ) = f(x 0 ), T n (x n ) = f (x n ) . Clearly, T n (x) is continuous on [0,1] no matter how we define T^x-t); i=l, ...,n-l. We shall uniquely determine the unknowns T n(Xi) by the conditions r n (3) (x i +) = T^ 3) (x i -) ; i = 1, ..., n 1. A simple computation gives, for i = 1, 2, ..., n

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107 [474^]] + T n (x Â± ) [1062 h\ + 1062^] + T a ( X Ul ) [474/2^] = fiz^) [1536*1] + f (zj) [1536^] + f'Ui.i) [-132A i _ 1 iii] + f'Ui) [426^21] 4262ii_ 1 2i i ] (4.2.22) + f / (x i+1 )[l32i2?. 1 i2 i ] + ^ /7 [-12il?_ 1 22i] + f" ^[-IShUhl IShUhj] + f // (x i+1 )[-l2i i 3 . 1 2jl] + f ' (4) (^.J 1 i,4 .3 Yo hi lhi + f (4) Oq) 3 l.4 l3 3 t 3 i_ 4 ~To bi lhi To hi ~ lhi + f U) U i+1 ) 20 1 1 1 Note that the equations in (4.2.22) make up a tridiagonal dominant system. Such a system has a unique solution. The unique spline T n (x) satisfying (4.2.21) and (4.2.22) provides the proof for Theorem 4.3. Proof of Theorem 4.4 The technique used to prove Theorem 4 . 2 can be applied here with only minor changes. Again, we define, for x = x* + thq, hi Â— x i+1 Â— Xi, (4.2.23) TÂ„(x) f (x) =11(0 +j li (t) where (4.2.24) Mt) = (T n ( Xl ) f(x i ))A 0 (t ) + (r n U 1+1 ) -f(x 1 . 1 ))A 0 ( 1-0 and MO = fiA, ( O + f in Ao (l t) + f iA x (0 2 + h x (f'iB 0 (t) f' itl B 0 (l-o ) + hi(f"c 0 ( O + f'^C Q (l-o) + ili(fi 4) Â£0 ( t) + flt[E 0 (l-t)j-f(x). (4.2.25)

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108 We may estimate ^(t) in exactly the same way as in the proof of Theorem 4.2. The estimate of A i (t) / however, is slightly different. After referring back to the tridiagonal dominant system in (4.2.22), we observe that l[ r n<*i-i) ~ fUj-i)] 474 hj + [T n {Xj) f (**)] [1062211 + 1062 (4.2.26) + [r n (x i+1 ) f (x i+1 )] 474hl_ 1 I Â£ |r n (x i ) f (Xj) I [588h] + 588AI.J when |T n (x i ) -f(x i )|= max \T n (x) f (x,) | . j=i-l, i, i+l J Now, we use (4.2.26) along with the technique from the proof of Theorem 4.2 to finish the estimate. Proof of Theorem 4.5 The proof of Theorem 4 . 5 follows in exactly the same way as the proof of Theorem 4.3. We, therefore, only list the equations which provide the unique explicit form for the spline Rn( x) e satisfying (4.1.7). Let x e [x t , x 1+1 ] . Then x = x t + th t where 0 t <; 1. R n (x) = f(x 1 )A 0 (t) +f(z i )A 1 (t) + f (x i+1 ) Aq (1 t) (4.2.27) + hiie'Ui) B 0 (t) + h Â± f' {zj B 1 (t) (x*^) B 0 (1 t) + h\f"' (x t ) D 0 (t) -hif'"(,x 1 ^)D 0 (i-t) for x e [x A , x i+1 ] where

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109 12Aq (t) = -6 00 t 7 + 2228 t 6 2946t 5 + 1475 t 4 + 169t 2 + 12 3^(0 = -64 1 6 + 192 t s 160 t 4 + 32 1 2 (4.2.28) 24B 0 (t) = -216 t 7 + 820 t 6 1122t s + 595t 4 101t 2 + 24t (t) = -32 1 7 + 112 t 6 136 t s + 6 0 1 4 4t 2 288D 0 (t) = 24 1 7 100t 6 + 162t 5 127 t 4 + 48t 3 7 t 2 And the unknowns T n (x 1 ) for i = 1, . n 1 are determined by the following tridiagonal dominant system of equations. For i = 1 , 2, . . . , n 1 , we have R'n Uj-i) [llh^hl] + Ri,{x 1 )[l0lh 1 . 1 h! + lOlhUhi] * R^x^llhUh^ = f (x 1 _ 1 )[-82hj] + f (Zi-J [-256h|] + f (x 1 )[338hf 338111-!] (4.2.29) + f (zj [25621*.!] + f (x i+1 )[ 82i3?_ x ] + f' (z^) [-96ll i _ 1 2lf] + f / (z i )[-96h?. 1 l3 i ] + f'"{x i ^) Proof of Theorem 4.6 . Along the same lines as the proof of Theorem 4.4.

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CHAPTER FIVE EXTREMAL PROPERTIES FOR THE DERIVATIVES OF ALGEBRAIC POLYNOMIALS Main Results In this chapter, we present three theorems which discuss the growth of the derivatives of algebraic polynomials bounded by curved majorants in the L p norm. Specifically, Theorems 5.1 and 5.2 state results in the L 2 norm. Note that Theorems 5.1 and 5.2 give the polynomials that possess derivatives with the maximum possible value for every n. Theorem 5.3 presents polynomials that possess derivatives of maximum value asymptotically. We now present our results . Throughout this chapter, T n (x) and u^x) are defined by T a ( x ) = cos n 0 and ( x ) sin n Q sin 0 where x = cos 0 Theorem 5 . 1 Let Pn+i ( x ) be a real algebraic polynomial of degree i n + 1 such that | p n+1 (x) | * (1 x 2 ) 2 , for -1 * x <; 1. Then, for k = 2, 3 Â• * Â• / we have i 7JC-1 -1 (5.1.1) i -l 110

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where f Q (x) = (1 x 2 ) u^x) . Equality iff p n+1 (x) = Â± fo(x) . Remark 1 . Ill The case k = 1, under the further assumption that the polynomial p n+ i(x) has all real zeros that lie inside [-1,1], is also treated in [44]. Next, we shall prove: Theorems 5 . 2 Let p n+2 (x) be a real algebraic polynomial of degree n + 2 such that \p n+2 ( x ) | Â£ 1 x 2 , for -1 Â£ x Â£ 1. Then for k = 3 , 4 , . . . ; we have where f x (x) = (1 x 2 ) T n (x) . Equality iff p n+2 (x) = Â± f^x) . Remark 2 : In the case k = 2 , we were not able to resolve the inequality (5.1.2) . Theorem 5 . 3 Let P n+2 (x) be an arbitrary polynomial of degree n + 2 such that | P n+2 (x) \ z 1 x 2 for -1 s x <; 1. Then i -l (5.1.2) i -l i (5.1.3)

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112 where p is any fixed positive integer and C p is a constant that depends on p. The preceding is best possible in the following sense. lim 1 n-Â°Â° n 2 p f \Pn +2 (X) \ 2p dx = 2 2p + l' where P nt2 (x) = (1 x 2 ) T n (x) . Lemmas Here we state and prove some lemmas which are needed in the proofs our theorems. We now state: Lemma 5 . 1 Let qn_i(x) be any algebraic polynomial of degree at most n 1 with real coefficients. Further, let __i (5.2.1) \g n _ 1 ( x ) | Â£ (1 -x 2 ) 2 , for -1 < x < 1. Then we have * (5.2.2) J [qn-iU)} 2 (1 x 2 ) 2 dx z -| {n 2 1) Equality iff q (x) = Â± s ^ n ^ , x = cos0 . Proof of this sin0 lemma is given in [45]. Lemma 5 . 2 Let c[n-i( x ) be any algebraic polynomial of degree n 1 with real coefficients such that |q n . 1 (x)| ^ (1 x 2 )' 1/2 f for -1 < x < 1. Then we have for k = 1, 2,

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113 (5.2.3) 1 2Jc+l J (x)] 2 (1 x 2 ) 2 dx <. / K3 -1 2ic+l (x) ] 2 (l x 2 ) 2 dx Equality iff q n _ x (x) = Â± u n _ x (x) ; (x) sin n 6 sin 0 ' x = cos 0 . Proof n-l We begin by setting g n _ 1 (x) = 2 jJ u 7 -(x) . J=0 Now using the orthogonal properties of {uj k) (x) } and { Uj (x) } , we obtain * 2Ar + l / [q^i (X)] 2 (1 X 2 ) 2 dx -1 (5.2.4) D-l 2k + l = s P 2 f f uj k) (x)] 2 (1 X 2 ) 2 dx j-k ^ L 1 and 1 3 1 1 3 (5.2.5) / [aU (x)] 2 (l-xV* = 2 f [u'j (x)] 2 (l x 2 )~ 2 dx -i Jml -i Next, we note that y = u^x) satisfies the differential equation (5.2.6) (1 x 2 ) y" 3 xy' + j (j + 2) y = 0 ,

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114 (5.2.7) From (5.2.6) it follows that (1 x 2 ) uj k) {x) {2k 1) xu J k ' 1) (x) + ( (j + l) 2 (k l) 2 ) uj k ' 2) (x) = 0 (5.2.8) Now on using integration by parts and (5.2.7), we have 1 2k *1 / [uj* 1 (x)] 2 (1 X 2 ) 2 dx -1 1 = -/ [uj (i> (x)](l X 2 ) 2k1 2 Â• [(1 x 2 ) Uj** 11 (x) (2k + 1) xuj k) (x)] dx 2k-l /[lij*" 1 Â’ (x)] 2 (1 X 2 ) 2 dx n ; [(j + l ) 2 k 2 ] Through repeated application of (5.2.8), we have 1 2k*l j [uj k) (X)] 2 (1 X 2 ) 2 dx (5.2.9) n ( (j + i ) 2 i 2 ) 1=2 f [uj (x)] 2 (1 x 2 ) 2 dx for k = 1, 2, ... where we define for k = 1, II ( (j + 1) 2 i 2 ) iÂ‘2 Sl .

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115 Hence, / [a<-i (x)] 2 (l x 2 ) -1 2Je + 1 2 23-1 = s j-Jc n ((j + 1 ) 2 i 2 ) i-2 n (n 2 i 2 ) i=2 n (i 2 2 i 2 ) i* 2 n (j? 2 i 2 ) i-2 n (n 2 i 2 ) i-2 Pj l [u'jix)] 2 (1 X 2 ) 2 dx -1 Â“s | [uj (x) ] 2 (1 x 2 )^ dx V p 2 / [uj(x)] 2 (1 x 2 )"* dx 2 _3 J [qn-i (x)] 2 (1 X 2 ) 2 dx -1 J [ui-i (x)] 2 (1 x 2 ) 2 dx -1 2 J :+1 = J [u n Â‘^ (x)] 2 (1 X 2 ) 2 dx Equality, iff g fl _ 1 (x) = Â± u D . x (x) . This completes the proof of Lemma 5.2. Lemma 5 . 3 Let q n _i(x) be any real algebraic polynomial of degree _i n 1 such that Ig^ (x) \ <> (1 x 2 ) 2 , for -1 < x < 1. Then, we have for k = 0, 1, ... (5.2.10) j [g^ (x) ] 2 (1 x 2 ) 2 dx \$ j [u n ( -i (x)] 2 (1 x 2 ) 2 dx -1 Equality holds iff q n . 1 (x) = Â± u^x) .

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Proof 116 Let v lf v 2 , . .., v n _* be the zeros of r n (Jc) (x) . Then (5.2.11) \g^l (v i ) | s \Un-l (v^) | for i = 1, 2, . .., n k For the proof of the above statement we refer to [45]. Now, using Gaussian quadrature formula, based on v lf v 2 , Â•Â•Â•/ v n _fc, we obtain / [ol-i (x) f (1 X 2 ) Â—*~dx = Ji [g^ k l (v^)] 2 -i Â• L_1 where K k) u )] 2 (x-vj) 2 [r n (i+1) (v*)] 2 2Jc-l (1 x 2 ) 2 dx z 0 fÂ° r i 1/ 2, ..., n k. In view of (5.2.11), we have * 2-fc-l nÂ— 1c f [g^ (x)] 2 (1 x 2 ) 2 dx <; 2 (v,)! 2 M d i=l L J ^ 2Jc-l = /[u n ( -i (x)f(l x 2 )^ dx -1 This completes the proof of Lemma 5.3. The previous three lemmas are needed for the proof of Theorem 5.1, while the subsequent three lemmas are used in the proof of Theorem 5.2. Lemma 5 . 4 l je ^ : < 3n-i( x ) be any real algebraic polynomial of degree at most n 1 such that Ig^ (x) \ z 1 for -1 s x s 1.

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117 Then we have: * i (5.2.12) f [q' D _ x (x)] 2 (1 x 2 ) 2 dx
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Following as in (5.2.8), we obtain (5.2.16) 1 2Jc-l /[r/" U )] 2 (l X 2 ) ~ dx Jc-1 n i1 1 / [rj (x) ] 2 (i x 2 ) 1 2 dx where for k Jc-1 1, we define II (j 2 i 2 ) = 1. i=l Hence, 1 2Jc-l / [q-j* 1 U)f (1 x 2 )~ dx = 2 J-* n (i 2 i 2 ) i-l a 2 / [rj (x)] 2 (1 x 2 ) 2 dx -1 1 x <; Â‘n (n 2 i 2 ) 2^ a 2 j [tj (x)] 2 (l x 2 ) 2 dx Jc-l Jc-1 Â£ (n 2 i 2 ) jE a 2 | [rj (x)] 2 (1 x 2 ) 2 dx Â» Jc-l 2 _1 = n (n 2 i 2 ) J [q/ (x)] 2 (1 x 2 ) 2 dx -l s (n 2 i 2 ) | [ t' d (X)] 2 (1 x 2 ) ^ dx -1 = f (x)] 2 (i x 2 )^dx This completes the proof of Lemma 5.5. Lemma 5 . 6 i ,e h qn(x) be any real algebraic polynomial of degree such that | g n (x) | <; 1, for -1 * x <; 1.

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119 Then, we have for k = 1, 2, ... (5.2.17) f [qr n (Jcl u)] 2 (l x 2 ) ~dx z f [r n (W U)] 2 (l X 2 ) ^ dx -i -i Equality holds iff q n (x) = Â± T n (x) . Proof Let u lf u 2/ . .., u n . k+1 be the zeros of r a (Jc-1) (x) . Then (5.2.18) | Qa k) (Ui) | s | (u*) |, for i = 1, 2, ..., n k + 1 Equality possible for any i if q n (x) = Â± T n (x) . For the proof of the above statement we refer to [45], page 104, formula (2.7.1). Now, using Gaussian quadrature formula, based on u lf u 2 , ..., u n _ k+1 , we obtain / [ ( Ui )] 2 H, -1 i=1 = f [r n u> (x)] 2 (i x 2 ) ^ dx , (X) (x-u^ r n (jc) (u*) 2Jc-3 (1 x 2 ) 2 dx ^ 0

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120 From this Lemma 5 . 6 follows . The final lemma is used in the proof of Theorem 5.3. Lemma 5 . 7 Let t n (0) be a trigonometric polynomial of order n. For p and k fixed positive integers, we have the following result . u Â« (5.2.19) f [t'(0)] 2p sin*0d0 <; n 2 f [t' D (0) ] 2p ~ 2 sin*0d0 + ^ n 2p ~ 2 The above is best possible in the following manner. -h / [^(Â«r si " ,:ede Â“2 (1^) / [^<Â«r si "* ed6 o o when t n (0) = cos n (0 a). Proof Define * (5.2.20) I x = f [fc'(0)] 2p sin*0d0. Clearly, we may write (5.2.21) I x = I 2 + I 3 , where * I 2 = / [t' (0)] 2 ^ 2 [ (c' (6)) 2 t n (0) (0) ] sin* 0d0 and

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^3 sin^OdQ = f t''( 0 ) t a ( 0 ) [t'( 0)] 2p ~ 2 From integration by parts. (5.2.22) J 3 = ~f t'(0) {[t'(0)] 2p_1 sin*Â© O * (2 p 2) fc''(0) t a (0) [^(0)] 2p ' 3 sin*0 + kt n (0) [tn(0)] 2p Â’ 2 sin^Ocos 0} d0 From (5.2.22), we have (5.2.23) I 3 = -I 1 ( 2p 2) I 3 + E, where IT E = -k J t B ( 0 ) [tn( 0)] 2p_1 sin^Ocos 0 d 0 Hence, (5.2.24) 1, = ~ 1 J, + Â— Â— Â£. v ' 3 2p 1 1 2p-l From (5.2.21) and (5.2.24), we have (5.2.25) 2p J, = J, + Â— Â— E. 2p-l 1 2 2p-l Further , / [t'(0)f P sin*0d0 = 2|_1 j [tn (0) ]' 2p-2 (5.2.26) *[(ti(0)) 2 t a (0) ta (0) ] ain*0d0 f tÂ»(0) [t-Wf 1 sin^OcosOdÂ©

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122 We now state the following inequality. (5.2.27) |[t^(0)] 2 t''(0) t a (0)| s n 2 for |t a (0)|sl. Equality for every 0 if and only if t n (0) = cos n(0 o). We now apply (5.2.27) to the first and Bernstein's Inequality to the second term on the right hand side of (5.2.26). We obtain (5.2.28) ? [t'(0)] 2p sin*0de s n 2 f \t' n (0)| 2p ~ 2 sin^OdÂ© + M n 2 ^ 1 . o 2p { 2p The lemma follows . Proof of Theorems We now present the proofs of Theorems 5.1 5.3. Proof of Theorem 5 . 1 We let p n+1 (x) be any real algebraic polynomial of _i degree n + 1 such that |p fl+1 (x) | s (1 x 2 ) 2 , for -1 * x * 1. Now, we write (5.3.1) p D+ 1 (x) = (1 x 2 ) g n _ 1 (x) where q^^x) is a real algebraic polynomial of degree n 1. Further we have, (5.3.2) \q n _ x (x) | * (1 x 2 ) 3 , for -1 < x < 1. Through repeated differentiation of (5.3.1), we obtain (5.3.3) pj" (x) = (1 x 2 ) (x) 2kxq^ k ~ 1] (x) (Jr l) kqÂ£ a) (x) , for k = 0, 1,2, ...

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123 From (5.3.3), we have 2Jc-3 (5.3.4) J [Pnii ( X ) ] 2 (l X 2 ) 2 dx = + l 2 + j 3 + j 4 + J 5 + I ( where and 2k*l J 1 = J (-X)] 2 (1 Â“ X 2 ) 2 dX, -1 2JC-3 (5.3.5) I 2 = Ak 2 f [g D ( _t 1) (x)] 2 x 2 (1 x 2 ) 2 dx, -l 2Jc-3 I 2 = (k l) 2 k 2 f[qÂ£ 2) (x)f (1 x 2 ) 2 dx, -l 2Jc-l J 4 = -Ak f [qj* (x) qÂ£ 1] (x)] x(l x 2 ) 2 dx -l Upon integration by parts 2Jc-3 I* = 2k f [Â«Â£Â» (x)f (1 x 2 ) 2 (1 2kx 2 ) dx. -1 ^ 2Jt-3 T s = 4 (Jr l) k 2 f [go 1 *' 11 (x)
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124 Similarly, we obtain ( 5 . 3 . 7 ) X 5 = -2 (Jc 1) Jc 2 f [g^ 2) (x) f 2k-S Â• (1 x 2 ) 2 [l 2 (Jc 1) x 2 ] dx Next, J 6 = -2 (Jc l)k f (x) qÂ£: 2) (x)] (1 x 2 ) 2 dx -1 1 2Jt-l = 2 (Jc 1) k f [^ u (x)f (1 x 2 ) ~ dx -1 1 2 (2Jc 1) (Jc 1) k J [gj.^ 11 (x) qn'-i 2 Â’ (x)] x(l > Note that i / [o ^ 15 U) qÂ±-: 2) (x)]x(l X 2 ) 2 dx -1 * 1 2k-5 = f [gÂ£* ] U)f (l x 2 ) Â“ [l 2 (k -i Hence, 2J:-1 ( 5 . 3 . 8 ) I 6 = 2 (Jc 1) Jc j [g^ 1 * (x)] 2 (1 x 2 ) 2 dx Â“1 1 + (2 Jc 1) (Jc 1 )k j [g^' 2 Â’ (x)] 2 2Jr-5 Â• (1 x 2 ) 2 [1 2 (Jc 1) X 2 ] dx 2k-3 ) 2 dx 1) x 2 ] dx

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125 (5.3 and From (5.3.4) (5.3.8), we obtain 2k-2 2k+l / [Pn*i U) f (1 x 2 ) 2 dx = J [g^ (x) f (1 x 2 ) 2 dx Â“1 -1 1 2k1 + 2 (Jc 1) k f [gj-t 11 (x)] 2 (l x 2 ) ~ dx -1 9 ) + 2k f (x)] 2 (1 x 2 )^ dx -1 1 2k-2 + (Jc 2) (Jc l) 2 Jc I [qlÂ™ [x)] 2 (1 x 2 ) ~ dx -1 + (21c 3) (Jt 1) Jc f [g^' 2 Â’ (x)f (l x 2 ) ^ dx . Finally, we use Lemmas 5.2 and 5.3 along with (5.3.2) (5.3.9) to get 2Jc-3 1 / [Pn*l U)f (1 X 2 ) 2 dx z f [uÂ» (x)] 2 (1 X 2 ) 2 dx -1 -1 + 2 (Jc Â§ 1) k f [u^11 (x)] 2 (1 X 2 ) ^ dx -1 \ 2*-3 + 2 Jc j [uj-V* (X)] 2 (1 X 2 ) 2 dx -1 + (J: 2) (Jc 1 ) 2 k I [uÂ£ 2) (x)] 2 (1 x 2 ) ^ dx -1 1 2k-5 + (2Jc 3) (Jc l) k j [u n < * 1 ' 2) (x)] 2 (1 x 2 ) ~ dx -1 = f [. f (x)] 2 (1 x 2 )^dx

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126 where f Q (x) = (1 x 2 )u n _ 1 (x) . Equality iff p n+1 (x) = Â± f 0 (x) . Proof of Theorem 5.1 is complete . Proof of Theorem 5.2 We let p n+2 (x) be any real algebraic polynomial of degree n + 2 such that |p n+2 ( x ) | s (1 x 2 ) , for -1 Â£ x Â£ 1. Now we write, (5.3.10) Pn+ 2 ( x ) = (1 x 2 ) q n (x) where q n (x) is a real algebraic polynomial of degree n. Further, we have (5.3.11) \g n (x) | Â£ 1 , for -1 * x s 1. Through repeated differentiation of (5.3.10), we get (5.3.12) pW (X) = (l X 2 ) qr n (Jc> (X) 2kxg n (Jc Â‘ 1> (x) (k 1) k g n (jc ' 2> (x) for k = 0, 1,2, ... From (5.3.12), we have (5.3.13) j, [p n ( " (x)] 2 (l x 2 ) ~~ 2 ~ dx = X! + J 2 + x 3 + j 4 + j 5 + j s -1 where Â• i 5 1c 1 -1 (5.3.14) -l -l

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127 and ^ 2Jc-3 J 4 = -Ak J [g a !Jc) (x) q^1] (x)] x(l x 2 ) 2 dx. -i Using integration by parts, (5.3.15) x 4 = 2k f [g^ 11 (x)f (1 x 2 ) ^ [1 2 (* 1) x 2 ] dx -1 J 5 = 4 (Jr 1) k 2 f [g n (jc_2) (x) gj*1 Â’ (x)] x(l x 2 ) ^ dx -1 Similarly, we obtain J s = -2 (it l) k 2 f [g n (jc ~ 2> (x)f (5.3.16) 1 2JC-7 Â• (1 x 2 ) 2 [l 2 (ic 2) x 2 ] dx Next, X 6 = -2 (Jt-l)Jt/ [g a (Jr ' 2> (x) g fl (w (x)] (1 x 2 ) ^ dx -1 2 (k 1) k f [g n (Jc_1> (x)f (1 X 2 ) ^ dx -1 ^ 2Jc-5 2(2k 3) (k l)k f [g n (Jc-1> (x) g n u ~ 2> (x)] x(l x 2 ) ~dx

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128 Observe that 2Jc-5 / [q n ( *1> ( x ) g D 1 *' 21 (x)] x(l x 2 ) 2 dx = f [dn k ' 2) U)f (1 x 2 ) [1 -2 (Jc 2) x 2 ] dx Hence, (5.3.17) 2 k2 J 6 = 2 (k 1) k j [g n ( *' 1> (x)] 2 (1 x 2 ) 2 dx -1 1 + (2Jc -3) (k-l)kf [q*2) (x)f (1 x 2 ) ^ [1-2 (Jc 2k-l From (5.3.13) (5.3.17), we may write (5.3.18) 2Je-S 1 / [?Â„<*> (Jc)] 2 (1 X 2 ) Â‘ V dx = / [g a l *> (X)] 2 (1 X 2 ) Â“ 1 dx -i -i ^ 2iCÂ”3 + 2 (Jc 3) Jc f [gj*' 11 (x)] 2 (1 x 2 ) ~ dx -1 * 2 *Â“ 5 + 6 k j [g^' 1 Â’ (x)] 2 (1 x 2 ) 2 dx -1 + (Jc 4) (Jc 3) (Jc 1) k [ [g n (Jc " 2) (x)] 2 (1 x 2 ) ^ dx -1 + 3 (2Jc 5) (Jc 1) k [ [g n ( *2) (x)] 2 (1 x 2 ) ^ dx 2 ) x 2 ] dx .

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129 Finally, we use Lemmas 5.5 and 5.6 along with (5.3.11) and (5.3.18) to obtain \ 2JC-5 \ Zk-l / [pÂ£l U)f (1 X 2 ) 2 dx <. [ [T n (W (x)] 2 (1 X 2 ) 2 dx -1 -1 1 2ic-3 + 2 (* 3) k f [rj*Â’ 11 (x)] 2 (1 x 2 ) ~ dx -1 1 2k-5 + 6 k J [r n 1) (x)] 2 (1 X 2 ) 2 dx -1 1 2 k5 + u 4) (it 3) [k 1) k J [r n (Jc ' 2> (x)] 2 (1 x 2 ) ~ dx -1 1 2k-l * 3 (2k 5) (k 1) k f [r 0 (jc ~ 2> (X )] 2 (1 x 2 ) ~ dx -1 = f [A U> U)f (1 x 2 )^dx . Â“I where f^x) = (1 x 2 ) T n (x) . Equality iff p n+2 = Â± f x (x) . Proof of Theorem 5 . 3 Define^ f Q (x) such that P n+2 (x) = (1 x 2 )f 0 (x). Then f Q (x) is a real algebraic polynomial of degree n such that \f Q (x) | <. 1 for -1 s x s 1. We now define (5.3.19) t n ( 0 ) = f Q (cos 0) for o ^ 0 ^ n. Then t n (0) is a trigonometric polynomial of order n such that (5.3.20) |t n (0)| s l for o ^ 0 ^ it.

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From the binomial theorem and Bernstein's Inequality (5.3.21) 1 it j |-Pn -2 (*) | 2p dx = J [sin0 t' n (0) + 2 cos0t n (0)] 2p sin 0 d0 -l o = ( 2p ) f [sin0 t' a (0) ] 2p sin0d0 + ( 2p ) O It Â• J [sin0t' (0)] 2 *' 1 [2 cos 0 t a (0)] 1 sin0d0 O -Â•Â•*(19/ [2 cos0t n (0)] 2p sin0d0 s J"[fca(0)] 2p sin 2p+1 0d0 O O + (Y) (2) 1 it n 2 ^ 1 + ( 2p ) (2) 2 n n 2p ~ 2 + ... + ( 2p ) (2) 2p it nÂ° . After repeated applications of Lemma 5.7, we have f [t'(0)] 2p sin 2p+1 0d0 O (5.3.22) (I) (I) (I) / 2p \ 2p ij | sin 2p+1 0d0 n 2P '+ (2p + l) it [ n 2p ~ 3 + n 2p ~ 3 + 2 [ p p-1 We conclude that (5.3.23) / |Pn +2 U) 1 2p dx <. (i) (t) (!)Â•Â• (^fr) / 6de 23 2p + (2p+l) 71 [ n 2 *1 + n 2p ~ 3 + + nf + pi 2 L p p-1 2 lj + ( 2p ) 2 m 2 p3 + ( 2p j 2 2 7tn 2p 2 + ... + j 2p ) 2 2p it n Â° . The theorem now follows.

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CHAPTER 6 SUMMARY AND CONCLUSIONS The problems addressed in this dissertation are taken from the vast field of approximation theory. The three areas considered are Birkhoff interpolation, lacunary spline interpolation, and Markov-type inequalities . In Chapters Two and Three, we discussed the explicit representation and convergence properties for a Birkhoff interpolation process. In Chapter Two, we found the unique polynomial of minimal degree that takes function values on one set of knots while its second derivative takes the value zero on another set of knots. Specifically, function values are interpolated at the zeros of (1 x 2 )P n _ 1 (x), and the second derivative values are prescribed to be zero at the zeros of p' Qr 1 ( x ) . Chapter Three gives a pointwise estimate for the error in using these polynomials to approximate a continuous function on the interval [-1,1]. The error is shown to be of order co 't, & n for -1 ^ x <; 1 We note that we have given a discrete interpolatory proof of the Tel jakovskii Theorem (Theorem 1.5). 131

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132 Chapter Four discussed (0,1,3) and (0,1, 2, 4) lacunary spline interpolation. Here, we allowed some of the data to be prescribed at the midpoints of the joints as well as at the joints of the spline. The splines constructed converge to an interpolated function with the optimal order of convergence for splines of their class. One of the (0,1, 2, 4) splines presented turns out to be a locallydefined process. In Chapter Five, we obtained various upper bounds for the derivative of polynomials under curved majorants in the L p norm. Here, p is an even integer and the majorants considered are the circular and parabolic ones. We conclude this section with some open problems inspired by the discussion presented in this work. In Chapter Three, one could improve the pointwise error estimate given in Theorem 3.1 by showing the result holds for the second modulus of continuity. This would provide a discrete, interpolatory proof of the DeVore Theorem (Theorem 1 . 6 ) . Another related problem would be to obtain the explicit forms and convergence properties for a similar Pal-type (0;2) interpolation process involving the zeros of the Tchebycheff polynomials of the first and second kind. Do such polynomials converge for the entire class of continuous functions? What sort of pointwise estimation of error is possible?

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133 In Theorem 5.3, we present an upper bound for the first derivative of polynomials under the parabolic majorant in the L p norm, where p is an even integer. Our upper bound is shown to be asymptotically sharp as discussed in (5.1.3). One could improve this result by giving an upper bound that is sharp for every n. In addition, one might ask what types of upper bounds are possible for higher derivatives.

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REFERENCES 1. Akhlaghi, M.R., Chak, A.M. , and Sharma, A., "(0, 3) interpolation on the zeros of u n (x)," Rocky Mtn. J. Math . 19 (Winter 1989), 9-37. 2. Akhlaghi, M.R. and Sharma, A., "Some Pal type interpolation problems , " Aprox . , Optimiz, and Comp; Theory and Applic . (1990), 37 40. 3. Bal&zs, J. and Tur&n, P., "Notes on interpolation II," Acta Math. Acad. Sci. Hung. 8 (1957), 201 215. 4. Bal&zs, J. and Tur&n, P., "Notes on interpolation III," Acta Math. Acad. Sci. Hung . 9 (1958), 195 214. 5. Bal&zs, J. and Turan, P., "Notes on interpolation IV," Acta Math. Acad. Sci. Hung . 9 (1958), 243 258. 6. Birkhoff, G.D., "General mean value and remainder theorems with applications to mechanical differentiation and integration, " Trans . Amer . Math . Soc . 7 ( 1906) , 107 136. 7. Bojanov, B.D., "An extension of a Markov inequality," J. of Approx. Theory . 35 (1982), 182 190. 8. Calderon, A.P. and Klein, G. , "On an extremum problem concerning trigonometric polynomials," Studia Math. 12 (1952), 166 169. 9. Demko, S., "Lacunary polynomial spline interpolation," SIAM J. Num. Anal. 13 (1976), 369 381. 10. DeVore, R.A., "Degree of approximation," Approximation Theory II , edited by Lorentz, Chui, and Schumaker (Academic Press 1976), New York, New York, 117 162. 11. Egervary, E. and Turan, P., "Notes on interpolation V, " Acta Math. Acad. Sci. Hung. 9 (1958), 259 267. 12. Erdos, P., "On the maximum of the fundamental functions of the ultrasperical polynomials." Annals of Math. (1944), 335 339. 134

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135 13. Fawzy, T., "Notes on lacunary interpolation by splines I , " Annales Univ. Soc . Budapest, Sectio Math. 2 8 (1985) , 17 28. 14. Fawzy, T., " Notes on lacunary interpolation by splines II, " Annales Univ. Sci. Budapest. Sectio Comp. . 29 (1986), 117 123. 15. Fawzy, T., "Notes on lacunary interpolation by splines III, " Acta Math. Hung. 50 (1987), 33 37. 16. Fawzy, T. and Schumaker, L.L., "A piecewise polynomial lacunary interpolation method, " J . Approx . Theory 48 (1986) , 407 426. 17. Fej6r, L., "Uber interpolation," Gottinqer Nachrichten . 66 91, Akademic der Wissenschaften, Gottingen (1916). 18. Freud, G., "Bemerkung uber die Konvergenz eines interpolations fahrens von P. Tur&n, " Acta Math. Sci. Hung. 9 (1958), 337 341. 19. Hill, E., Szego, G., and Tamerkin, J.D., "On some generalizations of a theorem of A. A. Markov, " Duke Math J. 3 (1937), 729 739. 20. Howell, G. and Varma, A.K., "(0, 2) interpolation with quartic splines," Numer. Funct . Anal, and Qptimiz . 11 (1990-91), 929 936. 21. Jackson, D., The Theory of Approximation . Amer. Math. Assoc., New York (1930). 22. LorentÂ’z, G.G., Birkhoff Interpolation . Addison-Wesley, Reading, Mass. (1983). 23. Meir, A. and Sharma, A., "Lacunary interpolation by splines," SIAM J. Num. Anal. 10 (1973), 433 442. 24. Mills, T.A. and Varma, A.K., "A new proof of Tel jakovskii's approximation theorem," Studia Sci. Math. Hung. 14 (1979), 241 256. 25. Pierre, R. and Rahman, Q.I., "On a problem of Turan about polynomials II," Conrad J. Math. 33 (1981), 701 733. 26. Pierre, R. and Rahman, Q.I., "On a problem of Turan about polynomials III," Conrad J. Math .. 34 (1982), 888 899.

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136 27. Pierre, R., Rahman, Q.I., and Schmeisser, G., "On polynomials with curved majorants," J . Approx . Theory 57 (1989), 211 222. 28. Pras&d, J. and Varma, A.K., "Lacunary interpolation by quintic splines," SIAM J. Num . Anal. 16 (1979), 1075 1079 . 29. Rahman, Q.I., "On a problem of Turan about polynomials with curved majorants," Trans. Amer. Math. Soc . 163 (1972) , 447 455. 30. Rahman, Q.I. and Schmeisser, G., "Markov-Duff inSchaeffer inequality for polynomials with a circular major ant," Trans. Amer. Math. Soc. 310 (1988), 693 702. 31. Rahman, Q.I., and Watt, A.O., "Polynomials with a parabolic majorant and the Duff in-Schaeffer inequality," J . Approx Theory 69 (1992), 338 354. 32. Sansone, G., Orthogonal Functions , Interscience Publishers Inc., Vol. 9, New York (1959). 33. Saxena, R.B. and Sharma, A., "Convergence of interpolatory polynomials, (0, 1, 3) interpolation," Acta Math. Acad. Sci. Hung. 9 (1958), 157 175. 34. Surdnyi, J. and Turan, P., "Notes on interpolation I," Acta Math. Acad. Sci. Hung. 6 (1955), 67 79. 35. Szabodos, J. and Varma, A.K., "On a convergent P&l type (0, 2) interpolation process," completed 1992. 36. Szabodos, J. and Varma, A.K., "On convergent (0, 3) interpolation processes," submitted Rocky Mtn. J. Math . 37. Szabodos, J. and Vertesi, P., Interpolation of Functions . World Scientific (1988). 38. Szego, G. , Orthogonal Polynomials . Amer. Math. Soc. Colloquium Pub., Vol. 23 (1959). 39. Timan, A., Theory of Approximation of Functions of a Real Variable , Pergamon Press, New York (1963). 40. Tur&n, P., "On some problems of approximation theory," J. Approx. Theory 29 (1980), 23 85. Varma, A.K., "A problem of P. Turan," Canad. Math. Bulletin 10 (1967), 531 557. 41.

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137 42. Varma, A.K., "Lacunary interpolation by splines I," Acta Math. Acad. Sci. Hung. 3 (1978), 185 192. 43. Varma, A.K., "Lacunary interpolation by splines II," Acta Math. Acad. Sci. Hung. 3 (1978), 193 203. 44. Varma, A.K., "Markov type inequalities for curved major ants in L 2 norm, " Colloquia Math. Soc . J&nos Bolyai 58 (1990), 689 697. 45. Varma, A.K., "Markov type inequalities for curved majorants in L 2 norm II," Aecruationes Mathematicae , to appear in 1993. 46. Varma, A.K., Mills, T.M. , and Smith, S., "Markov type inequalities for curved majorants," accepted Austral . J. Math. . 47. Varma, A.K. and Prasad, J. , "An analogue of a problem of J. Balazs and P. Turan, " Canad. J. Math. 21 (1969), 54 63. 48. Varma, A.K. and Yu, X.M., "Pointwise estimates for an interpolation process of S. N. Bernstein," J. Austral. Math. Soc. 51 (1991), 284 299. 49. Zygmund, A., Trigonometric Series . Cambridge University Press, London I and II (1959).

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BIOGRAPHICAL SKETCH John C. Burkett was born in Hagarstown, Maryland, in 1966. He completed the Bachelor of Science degree at Palm Beach Atlantic College during December, 1987. In January, 1988, he entered the University of Florida as a graduate student and teaching assistant. He received the Master of Science degree in December, 1989. 138

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (\yvpv Arun K. Varma, Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bruce H. Edwards Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Â• Bernard Mair Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M. Ij . ) cyiO&d' Popov Vasile M. Professor of Mathematics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Oceanographic Engineering December 1992