Some problems in approximation theory

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SOME PROBLEMS IN APPROXIMATION THEORY


By

BRANDON UNDERHILL











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


1996

















This dissertation is dedicated to the memories of

Professor Arun Kumar Varma

and

Robert Patrick McGrath














ACKNOWLEDGEMENTS


I express my deep gratitude to the late Professor Arun Kumar Varma for all

of his encouragement and many hours of enthusiastic assistance and instruction in

the area of Approximation Theory which made this dissertation possible. I extend

special thanks to Professor J. Szabados for his kind assistance on a trip to Gainesville

and thereafter. Similarly, special thanks go out to Professor A. Sharma and Profes-

sor T.M. Mills for their suggestions and encouragement. I thank the members of

my supervisory committee for their helpful comments and suggestions, and I thank

Mrs. Arlene Williams for her expert assistance. I thank my friends for keeping

me distracted, and finally, I thank my parents for all of their encouragement and

support.














TABLE OF CONTENTS




ACKNOWLEDGEMENTS ............................ iii

ABSTRACT ... .... .... ... .. .. .. .. ... .. .. ... vi

CHAPTERS

1 INTRODUCTION ............ .................. 1

1.1 Approximation by Polynomials .......... ........ 1
1.2 Lagrange and Hermite-Fejer Interpolation ............. 5
1.3 Birkhoff and Birkhoff-Fej6r Interpolation ............. 10
1.4 Markov-type Inequalities .......... ............ 12

2 BIRKHOFF INTERPOLATION : (0,1,3,4) CASE ........... 16

2.1 Preliminaries.... ... .16
2.2 Existence and Uniqueness ........... .......... 18
2.3 Explicit Representation ........... ........... 21

3 CONVERGENCE RESULTS FOR A BIRKHOFF-FEJER OPERATOR 28

3.1 Preliminaries and Convergence Theorem. . ..... 28
3.2 Estimate of the Fundamental Polynomials of the Fourth Kind .30
3.3 Estimate of the Fundamental Polynomials of the Third Kind .39
3.4 Estimate of the Fundamental Polynomials Ci(x) and C,(x) .41
3.5 Estimate of the Fundamental Polynomials of the Second Kind 43
3.6 Estimate of the Fundamental Polynomials of the First Kind 45
3.7 Proof of the Convergence Theorem . ..... 47

4 ERDOS-TYPE INEQUALITIES ..................... 49

4.1 M ain Results . ... . 49
4.2 Some Lemmas ...................... ...... .. 50
4.3 Proofs of Theorems ........................ .... .. 55

5 TURAN-TYPE INEQUALITIES ..................... 62

5.1 Main Results .......................... 62
5.2 Some Identities ............................ 64
5.3 Proofs of Theorems ....................... 70








6 SUMMARY AND CONCLUSIONS ..................... 78

6.1 Synopsis . . . ... 78
6.2 Open Problems ............. ...... ......... 79

REFERENCES ..................... .............. 81

BIOGRAPHICAL SKETCH .................... ........ 85














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




SOME PROBLEMS IN APPROXIMATION THEORY

By

Brandon Underhill

May 1996


Chairman: Dr. Joseph Glover
Major Department: Mathematics

We begin by providing an historical background and some results concerning

polynomial approximation and interpolation. Next we consider Birkhoff, or lacunary,

interpolation and its development. Then we provide the historical basis and develop-

ment of Markov-type inequalities, and related best constant problems when the class

of polynomials is restricted in some way.

We first investigate the (0,1,3,4) case of Birkhoff interpolation where the

nodes of interpolation are the zeros of the integral of the Legendre polynomial. We

prove the existence and uniqueness in the 'modified' (0, 1,3, 4) case, and then provide

an explicit representation in this case. Next we prove that the 'modified' (0, 1,3,4)

Birkhoff-Fejer operator (based on the zeros of the integral of the Legendre polynomial)

converges uniformly for the entire class of continuous functions on [-1, 1]. This

provides only the second known case of such a Birkhoff-Fejer operator the first

being the (0,3) case, both 'modified' and 'pure', studied first by Akhlaghi, Chak








and A. Sharma who proved the existence and uniqueness and provided explicit forms

for the fundamental polynomials, and then by J. Szabados and A.K. Varma who

provided a new representation for the fundamental polynomials of the first kind, and

proved the convergence results.

Let L, denote the Lorentz class of nonnegative polynomials of degree n on

[-1, 1]. In 1940, P. Erdos proved a refinement of Markov's inequality for polynomials

with all real zeros which are outside (-1, 1). We extend the results of P. Erdos, P.

Erdos and A. K. Varma, and G. V. Milovanovi6 and M. S. Petkovi6 for P, E Ln in

the L2 norm with the ultraspherical weight w(x) = (1 x2)a, a > -1, and we extend

these results in a weighted L4 norm.

Let now H, be the set of all polynomials of degree n whose zeros are all

real and lie inside [-1, 1]. We provide the lower bound analogues to the Erd6s-type

inequalities for P, E H,, as well as extend the results of P. Turin and A. K. Varma

with an asymptotically sharp result in the LP norm for p an even integer.

We conclude with a summary of the results and note some related open prob-

lems.














CHAPTER 1
INTRODUCTION

1.1 Approximation by Polynomials

In 1715, the English mathematician Brook Taylor (1685-1731) published his

generalization of the Mean Value Theorem [46]. His method approximates a given

n-times differentiable function f by a polynomial P, of degree n in (x a) required

to satisfy the conditions

(1.1.1)
P.k)(a) = f(k)(a), k = 0,..., n.

These conditions yield the polynomial

P(x) = -k(x a)k
k=0
the so-called nth Taylor polynomial. Taylor used these polynomials to approximate

solutions to equations.

Let us denote by C[a, b] the class of continuous functions on the interval [a, b].

Later on we shall have use for the big '0' notation, whereby O(k) means less than or

equal to a positive constant times k. Let us denote by HI, the class of all algebraic

polynomials of degree at most n, and denote the uniform norm for f C[-1, 1]

IlfI = max If(x)l.
-1
Lagrange proved the following version of Taylor's Theorem with remainder.

Theorem 1.1 If f and its first n + 1 derivatives are continuous on an open interval

(c, d) and if x and a are points of (c, d), then

f (x) = P,(x) + f(( a)
(n + 1)!








where 0,+1 is some number between x and a.
Unfortunately, the Taylor polynomials require a function f to be n-times

differentiable, and still may yield poor approximations to f outside a very small
neighborhood of a, as some derivatives of f at a may be very large compared to f(a).
Also, they are not a very efficient way to approximate a function. For example, the
error in the approximation for f(x) = ex by the third Taylor polynomial P3 about

a = 0 on the interval [-1, 1] can be seen to be

lie P3(x)I I 0.0516,

where the error is not evenly distributed through the interval. As is typical of ap-
proximations by Taylor polynomials, the error is much smaller near the origin than
near the endpoints 1.
In 1885, K. Weierstrass [55] discovered a theorem that founded a theory of

the approximation of functions, which can be stated as follows.


Theorem 1.2 If f is in C[-1, 1], then there exists a sequence of polynomials P, such
that P, -> f uniformly on [-1, 1].


The so-called Bernstein polynomials

B(f,x) = (2knn)( (1 + x)k(l x)n-k k k!(n -k)!'
k=0

provide such a sequence. Observe that if f 0, then Bn(f,x) > 0. Thus, we say
that Bn is a positive operator. It turns out that if f 2 0, then B, is an element
of the Lorentz class of polynomials, which we shall investigate further in Chapter 4.
The Bernstein polynomials also have the nice property that for any given r-times
continuously differentiable function f, we have B'j)(f, x) -+ f()(x) (j = 0,...,r) uni-
formly on [-1, 1]. On the other hand, the convergence of the Bernstein polynomials








is generally very slow. For example, if we choose f(x) = 22, then

lim n[Bn(f,x) f(x)] = x(1 z),
n-+oo

so that for large values of n we have

BIIB(f,x) x211 =0(-1x(1 x))
(n
Thus, the error does not decrease rapidly with n, even though f is a very simple
function.
Let us digress for a moment to consider how Bernstein may have come up
with these polynomials. Bernstein knew probability theory, and probably reasoned
as follows. First, suppose that the probability of an event occurring is x, where 0 <
x < 1, and so the probability of the event not occurring is 1 Now, the probability
of the event occurring precisely k times in n attempts is given by )xk(l k)
It follows then that
n n
(n )xk(l )n- = (sum of probabilities) = 1.
k=O k=O
One may also observe

1= [(1- x) + ] = k ) l ) n-k
k=O
Suppose now f E C[0, 1] and x is chosen randomly in [0,1]. For a given
positive integer n, consider the set {f( ) : k = 0,... ,n}. If n is large enough, then
at least one of the numbers f( ) lies close to f(x). We want to find a weighted sum

Ew(k, x)f (-), w(k,)= 1,
k=O k=O
that yields a good approximation to f(x). It follows from the Law of Large Numbers
in probability theory that choosing the weights w(k,x) = ( )Xk(l x)n-k, the
polynomials

f ) k( X)n-k
k=0








converge uniformly to the function f on the interval [0, 1]. These are simply the
Bernstein polynomials on [0, 1]. Once Bernstein found these polynomials, he gave a
proof of uniform convergence without using the Law of Large Numbers.
As a tool to measure the rate of convergence, we now introduce the classical
modulus of smoothness (or continuity) of order s

w,( f,) = sup IA'fl,
0
where

ALf(x) = f(x + h) f(x h),

Af (x) = Ar -f(x + h) Arh-f(x h).

It has been shown by G.G. Lorentz [21] that the error in the case of the Bernstein
polynomials is

IB.(f)- fj = OW /,f ,

and this is the best possible.
As C[-1, 1] is a normed linear space, for a given f E C[-1, 1], there always
exists a polynomial of best approximation to f. In fact, for each n there exists a
unique polynomial p IIE such that


IIf P~Il < IIf pll for all p IIE .

Set E,(f) = IIf-p;II. It has been shown by D. Jackson [19] that E,(f) < 6wi (f, ).
Because it is in general very difficult to obtain the polynomial of best approximation

p,, one often considers the best least squares approximation. That is, one minimizes
lf p, IL2 where

If|L2 = (L w(x){(f(x))2dx) 1

and where w(x) 2 0 and f w(x)dx exists. These approximations are usually pretty
easy to compute, typically yielding a good uniform approximation to f, superior








to that given by the Taylor polynomials. For example, we have for the function

f(x) = ex that the least squares approximation r*(x) (w(x) 1) of degree 3 gives
the error

Hex r*(x)ll 0.0112.

The error for the polynomial of best approximation p*(x) of degree 3 here is

lie p;(x)lI 0.0055.

The weight function w(x) allows for different levels of importance to be given to

the error at different points in the interval. Later, in Chapters 4 and 5, we shall

investigate some inequalities for polynomials in such a weighted L2 norm.

1.2 Lagrange and Hermite-Feijr Interpolation

Let us now consider the following problem. Given n distinct nodes

-1 < < n-1 < <

and data yi,y2,... y,, find an algebraic polynomial of least degree whose graph

passes through the points (xk, yk) for k = 1,...,n. That is, find a polynomial, say

Ln-1, such that
(1.2.1)

Ln-i(xk) = yk, k=l ,...,n.

We denote the Kronecker delta function Sk, by

Sk 1, if v = k,
kv{ 0, if v k.

Now, if we could find polynomials (x) (v = 1,.. n) such that

(1.2.2)


t,,(xk) =k, k = 1,...,n,








then we could write

(1.2.3)
n
Ln-1(x) = yv(W),
v=i
and the polynomial Ln-1 satisfies (1.2.1). But,


ea(x) =



is a polynomial satisfying (1.2.2). Thus, we have shown the existence of a polynomial

Ln-1 E n-1 satisfying the conditions (1.2.1).

Suppose now that there exists another polynomial pn-1 E IIn- that also

satisfies (1.2.1). Then

Ln-l(Xk) Pn-(Xk) = 0, k = 1,...,n,

and Ln- (x) Pn- (x) is a polynomial of degree at most n 1 having n zeros. Then

L,-_(x) pn-(x) =- 0, or equivalently, Ln-1 = pn-1. Thus, there always exists a

unique polynomial of degree at most n -1 satisfying (1.2.1), and it is given by (1.2.3).

We have just derived the so-called Lagrange interpolation formula. The polynomials

,(x) are called the fundamental polynomials of Lagrange interpolation. Lagrange
was interested in using interpolation to exploit the information in astronomical tables,

and around 1790 he presented a paper to the Academy of Sciences in Berlin.
If we define w(x) = (x Xi)(x x2) -. (x x,), then the numerator of e,(x)

can be written as (2) Observe then that

W'(xh) = (xv X1) .. (X xo-1)(x X"+i) ... (X, Xn)

is the denominator of ,(x). That is, we can write

,(X) = '(x )
(X X,)W'(xV)








This is a more useful form of the fundamental polynomials of Lagrange interpolation.

Concerning the error in Lagrange interpolation, Cauchy showed that for a function

f that is n-times differentiable, we have

f(x)- L, (x) =- f n ( W

for some number ( E (-1,1). Notice that if f II,,1, then f(n) = 0, and we have

Ln-1 f. We say then that Ln-1 is a projection operator.

One might think it reasonable to expect that if a system of equally spaced

nodes are chosen, then for a continuous function f on the interval [-1, 1] we should

have Ln- -+ f uniformly on [-1, 1]. In 1901, C. Runge [30] presented his classical

example that this is not necessarily the case. Let Xk = -1 + (k for k = 1,...n,

and choose f(x) = Runge showed that the Lagrange interpolation polynomial

does not converge to the continuous function f on [-1, 1]. In fact, he showed that

for 0.72 < |xj < 1, we have

lim |Ln-i(f,x) = 0o.
n--oo

This result is rather disappointing, and in fact in 1914, Faber [13] published his result

that for every choice of nodes

nE \(X) > log n
\ ( \ > ---

Unfortunately, this means that for any system of nodes, one can always find a function

f such that Ln-l(f) becomes unbounded. That is, there is no universally effective

system of nodes. A proof of this result can be found in the book of Rivlin [29]. Let

us define now the Lebesgue constant
n
An-1 = max if(x)I.
-1 V=1
It is easy to see that

|f Lll|j En-i(/f)(1 + An-1) < 6(1 + A_,1)W (/, -)
\ n'








Thus, for given f E C[-1,1] and a given system of nodes, L-,_(f,x) -+ f(x) uni-

formly if An-Iw (f, ) -+ 0.

In looking for interpolation polynomials which are uniformly convergent for the

whole class C[-1, 1], L. Fej6r [14] considered the so-called Hermite [17] interpolation

polynomial H2n-i E 112n-1. Here, given data yl,...,y, and y',...,y', we require

that H2n-1 satisfies the following conditions

H2n-l(Zk) = Yk, Hn-(xk) = y', k =1,...,n.

Thus, on any set of real distinct nodes, the Hermite interpolation polynomial has the
form
n n
tt2n- x) = ZykAk(x) + >Ey[Bk(x)
k=l k=l
where the fundamental polynomials are given by

Ak(x) = e (x)[1 2 '(xk)(x xk)], Bk(x) = (x Xk)(), k = 1,...,n.

In the case that the nodes are the zeros of the Chebyshev polynomial T,(x), we have

that

Ak(x) = (x) (- >0,

and it follows that
n
SIAk(x)l= 1.
k=l
Fej6r showed that for any continuous function f on [-1, 1], the operator
n
R2n-l(f,x) = f(xk)Ak(x)
k=1
converges uniformly to f on [-1, 1]. Two proofs of this result, as well as a nice
discussion, are given in the paper of T.M. Mills [27]. Note that this operator has the

properties that


R2n-i(f, k) = f(xk) and R'n-1(f, k) = 0, k = 1,...,n.








Thus, when all the higher derivative information is set equal to 0, we refer to such a
polynomial as a Hermite-Fejdr operator. This was one of the first interpolatory proofs
of the Weierstrass Approximation Theorem. It has been shown that for f E C[-i, 1]
we have

IiH2.-1(f) f\ = o (f), l

and this is the best possible in the sense that for the function g(x) = Jxz we have

HIn-i(g2 ) l> c ( ,10 n).

One can generalize the Lagrange and Hermite interpolation problem to the
so-called general Hermite interpolation problem (or simply Hermite interpolation).
In this problem we seek a polynomial Hnm-i satisfying the mn conditions


Hmi(xk) = y j = 0,...,m- 1, k = 1...,n,

where the numbers yk are given data. Notice that when n = 1, Hnm- is just the

(m 1)t Taylor polynomial. The (0,1,..., m 1) Hermite interpolation problem
always has a unique solution Hm-i E n,,m-1 on any set of real distinct nodes.

Despite the positive result in the (0, 1) case of Hermite interpolation, J. Sz-
abados and A.K. Varma [42] showed that the Lebesgue constant in the (0, 1,2) case
for every choice of nodes has the property
n
A3,-1 = max IAo,3,k (X)l > clogn,
S k=l
where the Ao,3,k(x) are the fundamental polynomials of the first kind of (0,1,2) inter-
polation. Thus, one cannot obtain uniform convergence for the whole class C[-1, 1]
for the (0, 1,2) Hermite-Fejer operator for any choice of nodes.
Later, J. Szabados [40] showed that
n m-1
Hn_(f,x)= E y)A ),
k=l j=O








where the fundamental polynomials of the (j + 1)t3 kind Ajk(x) of (0, 1,..., m 1)

Hermite interpolation are given by

Ak ) = k Jk(X Xk)i+j, j = 0,...,m -1 k=l,...,n,
Ajk(X) = -j i! '
i=O
and he showed that for every choice of nodes

|Aj)lkl> .1 (-cn) if m j is odd,
Sif j is even.
k=l nJ
In particular, when m is odd, we have for any system of nodes
n
II| Ao,k(x) >clog n
k=l
so that there cannot be uniform convergence for the whole class C[-1, 1] for the

(0,1,..., m 1) Hermite-Fej6r operator (m odd) for any choice of nodes.

1.3 Birkhoff and Birkhoff-Feijr Interpolation

In 1906, G. D. Birkhoff [8] considered the interpolation problem where in-

formation is prescribed for higher derivatives which are not consecutive. In this
case, unlike Hermite interpolation, a unique solution does not always exist. The

(0, mi,..., m,_l) Birkhoff (or lacunary) interpolation problem consists of finding a
polynomial Qs, such that


QJ(xk) = y k= 1,...,n, j= 0,m,...,ms-1,

where the yj) are given data and 0,mi,...,m,-_ are not all consecutive integers.

We see that Q, must satisfy sn conditions, so that Qn E IIan_1. In general, it is very
difficult to find an explicit representation of these polynomials. In fact, it is usually

difficult to even determine when there exists a unique solution to this problem.

Let Pn(x) denote the nth Legendre polynomial normalized by Pn(1) = 1,

and define irx,() = (1 x2)PI(x). In 1955, J. Surinyi and P. Turn [39] began

studying the case of (0,2) interpolation on the zeros of ir,(x). They showed that








for n even, the (0,2) interpolation problem has a unique solution. Later J. Balizs

and P. Turn [5] provided explicit forms for the fundamental polynomials. They

also proved convergence results and Markov-type inequalities with these polynomials.

The condition of convergence in this case was later improved by G. Freud [15] and H.

Gonska [16], but P. VWrtesi [54] has shown that the process is not uniformly convergent

for all continuous functions, as the Lebesgue constant of this type of interpolation is

of order exactly O(n).

In 1958, R.B. Saxena and A. Sharma [34],[35] extended the results of Turin

to (0, 1,3) interpolation, and later Saxena [32] extended them to the (0, 1,2,4) case.

In 1962, Saxena [33] handled the case of 'modified' (0, 2) interpolation on the same

nodes. Here, by 'modified', we mean that instead of prescribing second derivative

information at the endpoints +1, we prescribe first derivative information there.

We note that in general, 'modified' cases are more easily handled, and lead to the

solutions in the 'pure' cases.

A. K. Varma, R.B. Saxena and A. Saxena [52] studied the case of 'modified'

(0, 1,4) interpolation (second derivatives instead of fourth derivatives are prescribed

at 1) on the above zeros, showing the Lebesgue constant to be O(log n). Thus,

they conjectured that this process cannot converge for the whole class of contin-

uous functions on [-1, 1]. Recently, A. Sharma et al. [38] have shown that the

Lebesgue constant for the modified (0, 2, 3) interpolation (first derivatives instead of

third derivatives are prescribed at 1) on these zeros is also O(log n).

In looking for Birkhoff interpolation procedures which converge uniformly for

all continuous functions on [-1, 1], J. Szabados and A. K. Varma [43] considered

higher order (0, M) interpolation. Specifically, they were able to show that the (0, 3)

Birkhoff-Fejer operator on the above zeros converges uniformly for all continuous







functions on [-1,1]. More precisely, they proved that for f E C[-1, 1], the polyno-
mial R,(x) E IzI2n- satisfying

R,(xk) = f(k), R`(xk) = k= 1,...,n

has the property that

If(x) R(x)ll =0 ( (f,o )

where the xk are the zeros of ir,(x) and w3(f, 6) is the third modulus of smoothness of
f. Akhlaghi, Chak and Sharma [2] had already proved the existence and uniqueness
and provided explicit forms for the fundamental polynomials for the (0, 3) case, as
well as the (0,2,3) case [1]. We note that the 'modified' (0,2), (0,3) and (0,2,3) cases
on the zeros of 7r, (x) have been generalized [38] to the 'modified' (0,...,r 2, r),
(0,..., r 3, r) and (0,..., r 3, r 1, r) cases, respectively.
Given the positive result in the (0,3) case (and the negative results of the
others previously mentioned) we turned to consider the situation where we have a
similarly 'balanced' process. In particular, using an alternative representation to that
given by Sharma et al. [38], we show that the Lebesgue constant for the 'modified'
(0, 1, 3,4) interpolation (second derivatives instead of fourth derivatives are prescribed
at 1) on the above zeros is 0(1). This enables us to prove the uniform convergence
of the 'modified' (0, 1,3, 4) Birkhoff-Fej6r operator for the whole class of continuous
functions on [-1, 1].

1.4 Markov-type Inequalities

In 1889, A.A. Markov [24] proved that for any polynomial P, of degree < n
on the interval [-1, 1] we have
(1.4.1)

11^p11 < n21llp11,

where equality holds only for P,(x) = cT,(x), where T,(x) is the nh Chebyshev
polynomial. The Russian chemist Mendeleev [26] had settled the question for the








case n = 2 in his studies of the specific gravity of a substance as a function of the

percentage of the dissolved substance.

In 1892, A.A. Markov's brother W.A. Markov extended the Markov inequality

to all higher derivatives (published in German [25] in 1916), providing an inequality

sharp for every n. In 1912, S.N. Bernstein [6] improved Markov's inequality by

providing the following pointwise estimate

(1.4.2)
IP'(x)l < l P 11.

Note that this provides a much sharper inequality, except near the endpoints 1.

Both the Markov and Bernstein inequalities play a key role in proving convergence

theorems, as we shall see later.

The inequalities of A.A. Markov and S.N. Bernstein can be improved if the

class of polynomials is restricted in some way. Let us denote by Sn the set of all

polynomials whose degree is n and whose zeros are all real and lie outside (-1, 1),

and denote by L, the set of all polynomials of the form

(1.4.3)
P,(x)= Eakqnk(x), ak >0 (k = 0, 1,..., n)
k=0
where qnk(X) = (1 + X)n-k(1 X)k. In 1940, P. Erd6s [10] proved the following

refinement of Markov's inequality.

Theorem 1.4.1 (P. Erdos, 1940) Let P, E Sn. Then we have
1
11P'11 < 21enj1P 1,
2-

where the constant 'e cannot be replaced by a smaller one.

In 1937, E. Hille, G. Szego and J. D. Tamarkin [18] had extended the Markov

inequality in the LP norm, showing there exists a constant A such that for every

algebraic polynomial Pn(x) of degree n we have
1 P
f p (x) P dx)- < An2 ( IPn(x)Pdx)-
\J-l \J-l








where p > 1 and A is independent of n and Pn(x). Further, they noted that the
problem of obtaining the best constant in the above problem is extremely difficult.
In 1986, P. Erd6s and A. K. Varma [11] settled the above inequality for the
Lorentz class Ln of polynomials in the L2 norm as follows.
Theorem 1.4.2 (P. Erd6s and A. K. Varma, 1986) Let P, E Ln, n > 2. Then we have
(1.4.4)
f'I < n(n 1)(2n + 1) (pf'
(P(x))d 4(2n (3) n))2d
with equality if and only if Pn(x) = c(1 x)n-1(1 T x).
Also, if P, E Ln, then we have
(1.4.5)

(1- x2) (P(x)) 2d < + 1)(2n + 3) (1 x2)(Pn(x))2dx
1 4(2n + 1) i

with equality if and only if P,(x) = c(1 x)n.
It is known [22] that if Pn E Sn, then Pn E L, or -P, E Ln. Thus, (1.4.4)
can be viewed as an extension of Theorem 1.4.1 in the L2 norm. In 1988, G. V.
Milovanovi6 and M.S. Petkovi6 [28] extended (1.4.5) with the ultraspherical weight
(1 x2)", a > 1 (a > -1 if P,(1) = 0). A new proof is provided, and the
requirement that P,(1) = 0 is removed. Then an extension is provided in a weighted
L4 norm.
Let now Hn denote the set of all polynomials of degree n whose zeros are all
real and lie inside [-1, 1]. In 1939, P. Turn [45] proved this analogue of Markov's
inequality.
Theorem 1.4.3 (P. Turn, 1939) Let Pn H,. Then we have

11P.111 > 6 11P.III

This result was later sharpened by J. Erod [12]. In 1983, A.K. Varma [51] extended
the above in the L2 norm as follows







Theorem 1.4.4 (A. K. Varma, 1983) Let P, E H, and n = 2m. Then we have

f'(P(x))2dx + 4 + +(n- 1) (P())

where equality holds if and only if P,(x) = c(1 x2)m.
Moreover, if n = 2m 1, then for n > 3 we have


(P2(x))2dx +- + 4(n 2)) 1(, ))2dx

where holds equality if and only if P,,(x) = (1 x)m(1 :F x)m-'.
Earlier, A.K. Varma had given asymptotically sharp versions [48],[49] of this
result, as well as proved the following [50].
Theorem 1.4.5 (A.K. Varma, 1979) Let Pn E H,. Then we have (n = 2m)

(1 x2)(P(x))2dx > + 1 _(1 (
S- 4(nl 1)+ (1 -- x2)(P,(x))2dx,

where equality holds if and only if P,(x) = c(1 X2)".
We extend this result in the ultraspherical weight w(x) = (1 x2)a, a > 1,
(a > -1 if P,,(1) = 0). By a result of S.P. Zhou [56], for Pn, E H, and 1 < p < oo
there exists a constant B, independent of n and Pn, such that

( IPn(x dx}p >BO I P dx) P

We provide an asymptotically sharp result for p an even integer, showing in the limit
that
B (p- 1)(p- 3).-- 53
p2














CHAPTER 2
BIRKHOFF INTERPOLATION : (0,1,3,4) CASE

2.1 Preliminaries

The objective of this chapter is prove that the problem of 'modified' (0,1,3,4)

interpolation on the zeros of the polynomial 7r,(x) has a unique solution, and to pro-

vide an explicit representation in this case. By 'modified', we mean that instead of

prescribing fourth derivatives at the endpoints +1, we prescribe second derivatives

there. First we take care of some preliminary items.


Let P,(x) denote the nth Legendre polynomial normalized by Pn(1) = 1. Let

(2.1.1)

7r,(x) = (1 x2)P'_,(x)

and enumerate the zeros of 7rn(x) by

(2.1.2)

-1 = Xn < Xn_- < ... < Xi= 1.

We list the following known identities. These can be found in the book of G. Sansone

[31].

(2.1.3)

[(1 x2)P_, (x)]' + n(n 1)P,_1(x) = 0

(2.1.4)

xP_1(x) Pn_2(x) = (n- 1)P-1(x)

(2.1.5)
(1 x2)P _(X) = (n 1)[Pn-2(x) xPn_(x)]








(2.1.6)


P'(x))- P_2(x) = (2n 1)Pn-l(x)


(2.1.7)


(1 -x2)7r(x) + n(n 1)rn(x) = 0


(2.1.8)


(x) (x 7r,( )
x x)7r' (x,)


(2.1.9)


I'( })
(x x )i^k)(x)+ ketk-l)x)^


and we make note of the following values


(2.1.10)


Sn(n- 1)
-_l(1) = n(n


S-n (n 1)2
2


r'(1) =


-n2(n 1)2(n + 1)(n 2)


Sn(n- 1)
4


(n + 1)n(n 1)(n 2)
= 24


1 (n + 2)(n + l)n(n 1)(n 2)(n- 3)
192


(2.1.13)


)X n(n 1)
S -5(1 x)2(n n +


Observe from (2.1.7) that 7r"(x,,) = 0,


n(n 1)
3(1 x)


24
18 1 ),



v= 2,...,n-1.


v= 2,...,n 1.


(2.1.11)


p(n + 1)n(n l)(n 2)
Pn- ) = 8


(1 )=-n(n 1)


(2.1.12)


(x,) = 0


'(n(n 1)x,
S- (1 x )2








2.2 Existence and Uniqueness

We shall prove the following.
Theorem 2.1 Let a,, b, d,,, v = 1,..., n; cl, cn, and e,, v = 2,..., n 1 be given real
numbers. Then there exists a unique polynomial Qn(x) of degree < 4n 1 such that


Q.(x,,) = a,,, Q'(x,) = by, V = 1,... ,n

Qn(1) = c1, Q"n(-1) = Cn,

(2.2.1)

Q'(zX,) = d,, = l,...,n

Q()(x,) = e,, = 2,...,n- 1.



From linear algebra, this is equivalent to proving that if Qn(x) is a polynomial of
degree < 4n 1 satisfying
(2.2.2)

Qn(X)= Q',(x) = Q'(x,) = 0, = 1,...,n

Q1(+I1) = 0,

Q(4)(x,) = O, I= 2,...,n- 1,


then Q, = 0.


Before we proceed with the proof, we note that for the polynomial rv2(x) we
have
(2.2.3)


7r (x,) = [7r?(x)]' = 0, [(r(x)] = 27r(x,)2, = 1,.. n,








[r 1in- 2 (4) -8n(n l)7r^(a; )2
[lx()] = 0, [7L(x)]L = 1--2 =2,...,n-1.




Proof of Existence and Uniqueness


Suppose that Qn H4,Il satisfies the conditions in (2.2.2). We show that
then Qn = 0. Since Qn(x,) = Q'(x,) = 0 for v = 1,...,n, we write Q(x) =
7r2(x)q2,_1i(), where q2n-1 E I12n-1. Then we have

Q(1) = r(1)q2n-_(1) + 2[ r(x)] 1q_.(-1) + [7r (x)]1q2n-1()

= [(x)]lq2n-l(1).


As Q"(1) = 0 and [7r'(x)]+ = 27r(11)2 : 0, we have q2n-1(l) = 0.

7r2 2n xl[7r2(X), [ -3rr2(Ir x (x) I" q2n- I (X.)

Q"1(X,) = 7 (x,)q"2Nn(,)+3[1 ()]'q-1'+ 3 ),Q 2n-1 v 92n)] -1 &
= 3[,-( (x)'mq2,_ (x,), v = 1,...,n,
3nd 2sincel~q, _J(X.,)
and since [r2(x)]" = 27r'(x,)2 7 0, we have q2n-1() = 0 for = 1,...,n. From

the paper of Aklaghi, Chak and Sharma [1], page 63, we have q2n-1 IIn-2, and
there exists sn-i E IIl such that
(2.2.4)
q2n-l(x) = 7rn(x)sn1(x) 7'(x)Sn-I(x)

where s,_i(l) = 0. Using (2.1.7), we have
(2.2.5)
Sn(n 1
1-4V








Thus,

qn-'(x,) = r(xX)s"_1(x.) 7r"(x,)S,-l(x,)


7r(x,,)
((1 x)s-(,) + n(n 1)n-()) .

Now,

() = 6[r(x)]q~~ n-_(x,,) + [r(x)]) q2n- l(x,)


12= () ((1 x)sI_(x) + n(n 1)sn-X,))

87r' (,)3
+ n(n 1)s, (x)

= 47 (3(1 x -)s" (x) + 5n(n l)s-1(x,)) = 0,

v =2,...,n- 1.

As 7r(x,,) 4 0, we have

(2.2.6)


3(1 X)sn_l(x,) + 5n(n 1)sn_l(xz) = 0, v = 2,..., n 1.


In fact, since sn-,(l) = 0, (2.2.6) holds for v = 1,..., n. Then


3(1 x2)s"_,(x) + 5n(n 1)sn-l(z)

is a polynomial of degree at most n 1 having n zeros. We conclude

(2.2.7)

3(1 x2)s"_1(s) + 5n(n 1)sn-(x) 0.

Since s,_l(l) = 0, we can write

(2.2.8)
n-l
Sn-i(x) = Z z,7*r=(x).
v=2








Replacing (2.2.8) in (2.2.7) yields
n-1
1[5n(n 1) 3v(v 1)]zr,(x) = 0
v=2
so that z, 0 for v = 2,..., n 1. Thus, s,_- = 0 and we deduce q2n-1 = 0. This


gives Qn 0. Theorem 2.1 follows.


2.3 Explicit Representation


The polynomial Q, in the Theorem 2.1 will evidently have the form


n n n n-1
Q.(x) = E a,A,(x) + b ,B,(x) + ciCi(x) + nCn,(x) + d,D,(x) + > eE,(x)
v=1 v=1 v=l v=2


where the uniquely determined polynomials A,(x), B,(x), D(x) (v = 1,...,n),


(v = 2,... ,n 1) of degree < 4n 1 are characterized by


the conditions

(2.3.1)


A,,(k) = 8ky, A',(k) = A'(Zk) =

A"(1) = A"(-1) = 0, A4) (k) = 0


3 (k= ,...,n)

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


B,(zk) = 0,


B'(Zk) = 6kv,


B k) = 0


(k = 1,...,n)


B"(1) = B'(-1) = 0,


B(4)(xk) = 0


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


Cl(xxk C(k) = C'(k) = 0


C'(-1) = 0,


C1 (k) = 0


(k= 2,...,n 1)


Cn(xk) = C'(xk) = C'(xk) =0 (k = 1,..., n)


(2.3.2)


(2.3.3)


C'(1) = 1,


(2.3.4)


Cl(x), Cn(x), E,(x)


(k = 1,..., n)








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


(2.3.5)


D,(xk) D(x D'k) = 0,

D"(1) = D"(-1) = 0,
Dv1 v


D4) (k) = 0


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


E,,(k) = E,(xk) = E,(Zk) = 0 (k = 1,...,n)


E"(1) = E(-1) = 0,


E4)(xk) = Sk,


(k = 2,...,n 1)


Theorem 2.2 The fundamental polynomials of the 'modified' (0, 1,3,4) interpolation
based on the zeros of the polynomial 7rn(x) can be explicitly represented in the fol-
lowing manner.
(2.3.7)


E, (x) r(x)(1 x) n-
-4n(n 1)P _((,,) k=2


(2k )P'-1(x,) 7(x)k() n(x(x))
k(k 1)Ak (


where Ak = 5n(n- 1) 3k(k- 1), k = 2,..., n 1,


(2.3.8)


D,(x) ,)3


4 n-1
X) E Pn,-(Xk)=2(Xk)Ek(X)


(v = 1,.. ., n)


C (x)e~(x) (1 x)wr(x)el(x)
C1(x) = 2 4n(n-)
2n2(n 1)2 4n(n 1)


n--
-12 Pn-(xk) (xk)2
k=2


3n(n 1)
2


n(n 1)
2 (- x- 1) Ekx)


Cn(x) = Ci(-x)


(2.3.6)


(2.3.9)


C )(Xk) = 0


C"(1) = 0,


C",(-1) = 1,


D" (k)= (k = ,..., n)








(2.3.10)

n,(_ ) ((x) n(n 1)(13n2 13n + 1)
Bi() (x 2n(n 1)C(x) n(n )(13n2 13n + 1D(x)
n(n 1) 8
n-1
-24 P_-l (Xk).i (xk)3Ek(x)
k=2


Bn(x) = -BI(-z)


(2.3.11)


BL -,(x) +
7rn (x.)


4n(n 1) D 13n(n )1z,
1- -- 22 E(2)
1 v( i


+ 4 n-1
Pn1(x(,) kE
(v = 2,...,n- 1)


(2.3.12)


Ai(x) = e(x) n(n 1)Bl(x) 4(3f'(1)2 + l(1))CI(x)
n-1
-4(6e+(1)3 + 91 (1)D'(1) + "(1))Di(x) 24 )(xk)4Ek(x)
k=2


An(x) = Ai(-x)


(2.3.13)


4n(n 1)xzvD
A,(x) = -(x) + )D,x(z) +
(1 x )I


n-1
-24 ,(xk)4Ek(x)
k=2


24n(n 1) (
5(1 x)2 n


(v = 2,...,n 1).


- n ) E,(x)
1 v








Proof of Theorem 2.2


We first provide the proof of the representation of the last fundamental poly-

nomials. Following the proof of Theorem 2.1, we write E,,(x) = i7r(x)q2n- i(x) where

q2n-i(x) = 7rn(x)s'_1(x 7r'(x)sn-1(x) E H2n-2. Then, instead of (2.2.6) we get

4r(x (3(1 X2 )-i(k) + 5n(n l)n-(k)) = k,, k = 2,..., n 1,
1 Xk

or equivalently,

1-z
1 a2
3(1 x )s"1(k) + 5n(n 1)Sn-1(Xk) = 4(X,(k), k= 1,...,n.
47ra(X,,)3



Hence,
1 Z ().
3(1 x2)s"-1() + 5n(n 1)s,_1(x) 4 ,(r().



Now, using the identity

1 2 1
,(x) = n(n -- 1)Pn ) L k 1 Pk-(X)( v = 2,...,n 1,

which can be found in the (0,2,3) paper of Akhlaghi, Chak and Sharma [1], page 58,

and using the representation (2.2.8) we obtain

n(n 1)(i x2)(2k 1)P_,(x,)
zk 4k(k 1)Akr(x)5

so that
n(n 1)(1 x2) 1 (2k 1)P_1(xX)
47r'(x2,)5 = k(k 1)Ak

and we have (2.3.7).
To verify (2.3.8) we observe (2.3.5), (2.3.6) and that for the polynomial 7rn(x),,(x)

we have


S(xk)4(Xk) = [ (),(x) = [7r(x),(x)]k = 0 k = 1,..., n
n \ kcXj- ~\/ C\ k k-Ll\lVu 2









['(),( )] = 6r (x)36k k = 1,...,n


47r(xk)3 f'2k)
[l (x)4(x)] = 3( )
7T-(.A)


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


To verify (2.3.9), we observe (2.3.3), (2.3.5), (2.3.6) and that for the polyno-


mial


2n(n 1)2
2n2(n 1)2


(1 x)n-(x)l(x)
4n(n 1)


2n2(n 1k)
2n2(n 1)2


+ (1 xk)r (xk) 2(xk)
4+ 0 ,
4n(n 1)


I (x)x) (1 x) (x)'
2n2(n- 1)2 4n(n- 1) J k 1

r(x)e~(x) (1 xr())]"= 1,
.2n2(n-1)2 4n(n 1) J1
2f() ((x) (1 z)7r(2) (x) "
S2n2(n -1)2 4n(n 1)r)"
2n2(n 1)2 4n(n 1) -1


2n2(n ()2
2n2(n 1)2


(1 x)7r(x)(x)]"
4n(n 1) Jk


2n2(n 1)2


n(1 n- ) (x)() (4
4n(n 1) 1k


127r (xk)2e (xk)2(1 -_ (1))
n2(n 1)2


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


we have


3n(n 1)
2


k = 1,..., n,








To verify (2.3.10) we observe (2.3.2)-(2.3.6) and that for the polynomial
7r,(tx)t3() we have


7r(xk)e(Xk) = 0,


[7M(X) 1(Xs)]' = 7r(1)6k1 = -n(n 1)Ski,


[7rn(x)(x)()]k = -2n2(n 1)2k,


n2(n 1)2(13n2 13n + 1)k
8


k= 1,...,n,


[n(X)f(X)1 = 24 (X)el(k k) ,


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


To verify (2.3.11) we observe (2.3.2), (2.3.5), (2.3.6) and that for the polyno-
mial 7rn(x)e~(x) we have

7n(xk)t(xk) = [7n(x)e~(x)]'k = 0, k = 1,...,n


4n(n 1)7r(x(,)
[7rnk(X).(X)K =J1 Sk


[7rn(x)tyi(x)] )= 24r,(xk),(xk)3, k =


[ (7r, )( 4) 13n(n 1)xzxr(x1,)
(1 )2


k :v,


rn(X)'(X)]'x =


[n(X) (X)];" = 7 (X,)Sk,








To verify (2.3.12), we observe (2.3.1)-(2.3.6) and that for the polynomial f(x)
we have


I(zk) = Jkl,


[e(X)l = 4Ve(1)Skl = n(n 1)Skl,


k = 1,...,n,


[(=(X)]" ( (12i(1)2 + 4l(1))sk1


n(n 1)(11n2 1n 4),
12


[f (x)].. = (24t'.(1)3 + 36f (1)'(1) + 4i"'(1))Sk1,


[t(x)]~1 = 24e1(xk)4,


Finally, to verify (2.3.13) we observe (2.3.1), (2.3.5), (2.3.6) and that


(xk) = Sk, [e(



[t(z)]i, = 0, [ ,(X)]" =



[j(x)] ) = 24',(xk)4,


)]k= 0, k = ,...n,


4n(n 1)x
(1- x~~ k k
(1 Sl)2


,24n(n -1) 4
[e4(x)] ) = 36'(,)2 +4 )(,, ) = 24n(n2- ) 2 2)
5(1 2)2 1 X


This completes the proof of Theorem 2.2.


k = 1,...,n,


k = 1,...,n,


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


k :v,


v = 2,...,n-1.













CHAPTER 3
CONVERGENCE RESULTS FOR A BIRKHOFF-FEJER OPERATOR

3.1 Preliminaries and Convergence Theorem

Let f be a real-valued function on the interval [-1, 1], and define the 'modified'

(0, 1,3, 4) operator

R.(f,x) = Ef(,)A,(x),
iv=1
where the fundamental polynomials of the first kind A,(x) (v = 1,..., n) are given

by (2.3.12)-(2.3.13).

The main goal of this chapter is to prove the following.

Theorem 3.1 Let f be a continuous function on the interval [-1, 1]. Then

IIf(x) Rn(f,x)|I = (wl(, log n,))

where wl (f, 6) is the first modulus of smoothness of f.


Observe that the rate of convergence is the same as that of the classical

Hermite-Fej6r operator, but it is not as good as that of the (0,3) Birkhoff-Fej&r

operator on the zeros of 7r,(x).



We shall employ the following notations
(3.1.1)

x=cost, k = cost (k=l,...,n)

(3.1.2)








We shall use the following known estimates.


(3.1.3)


( 1 2)


(1 x2)7)



Ir,(x)I = 0 (1 )i n) ,


These can be found in Szeg6's book [44].

(3.1.6)

IPn(x) + Pn+1(x) = O sin t
n 2


1
-1< X < -.
2--2


This is a result from the paper of Szabados and Varma [43], page 734. There exist

absolute positive constants a1, a2, a3, a4 such that

(3.1.7)
2 < 1 2 = 2
(n 1) x (n 1)2 2 2


(3.1.8)


(n 1V)2
(n 1)2


(3.1.9)


(n 1)2



p2 a3
-(X) > nsint


(3.1.10)


Itv tV+l1 > --
n


(3.1.11)


t+tv
sin t < sin t + sin t, < 2 sin -- ,
2
t+t/
sin t < sin t + sin t, 2 sin --
2


(3.1.4)


(3.1.5)


= [--1]+1,...,n-1,










We shall refer several times to using an Abel transformation with factors bk, by which
we mean that we are using the summation by parts formula
q q-1
E akbk = Ak(bk bk+1) + Aqbq + Aplbp,
k=p k=p
k
where Ak = as.
s=O




3.2 Estimate of the Fundamental Polynomials of the Fourth Kind

Lemma 3.2.1 For the fundamental polynomials E,,(x) (v = 2,..., n 1) we have the
following estimate

n6 SiE = O sin2 2 ) if t,I> >
0 sint in3 t sin4 tv \ _t\
IEo(x)I 0sin i n sin2 n ,
O if It- ti~ < ,
where c > 0 is an absolute constant.
Proof

We shall first prove the case when t t,1 > -. Using (2.3.7), (2.1.1) and (2.1.3) we
have

S(ixr(x)(1 x -) ,2) 7 ( 12k 1
E1,(x) 4n4(n 1)4P, ,) (1 x2)rk(k- 1)Ak k-1X)pk-X)

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

k=2


S1 + S2.

Following the same argument given in the paper of Szabados and Varma [43], pages
736-738, with Ak = 5n(n 1) 3k(k 1) we obtain

n-1 (2k- 1) P'l(,)P '-()P_-(x)
Sk=k(k 1)AXk n-, (x Xv)








S sin 2 + sin t,
+0 tt3 2 2
n sin t sin2 t,, sin2 i sin ]
2


Thus,


7r (x)(1 X')(1 X')r,(X)P,'_-(x)P-'_(x,,)
S1
S1 4n4(n 1)4Psl(x,)An-l(x x,)

2r(x)(i x2)(i x2)r,(x) ( sin_ ,
4n4(n 1)4 P5_3(2,) .O2
4n(n 1)- 1(x) n sin2 t sin t1, sin2 sin


The first term in S1 will later cancel with a term occurring in S2. Using (2.1.3),
(3.1.5), (3.1.9) and (3.1.11) we obtain the estimate


O ( sin3 t,,
n6 sin 112


sin4 t,
+n6 sin-2 v


for the remaining term in S1.
We turn now to estimate S2. Using an Abel transformation with the factors

yields
(3.2.1)
7 (X)(1 X2) "-1 2k 1
S 4 = E 2k Pk-1(x,)Pk-1 ()
4n(n 1)P() k=2 k

7r (X)(1 x) 1 n-2
n( ) ) I 1 (2k + 1)Pk(x)P'(x,)
4n4(n 1)4p5_i(x) A-1 k=1
^.^^;i^[


n-2 k=l
-6 2 (2s
k=2 1


+ 1)P8(x)P.,(x.))].-


Next, we differentiate both sides of the Christoffel-Darboux formula
(3.2.2)

(2r + 1)Pr(x)P(y) = (n + 1)P()P+() P+()P()
r=O y X










with respect to y and set z, equal to y to obtain
(3.2.3)
SP.(X) )P"'z(2) Pn+i()P'(x,,)
E(2r + 1)Pr(x)PF(X,) = (n + 1) P-+-(x)P-(x- )
r=O
P,(x)P.+1(x,) Pn+l(X)Pn(x,)]
(x5 J)2


and applying this result to (3.2.1) yields


n 1 Pn-2(x)P-(x,,) Pn-l(x)P _2(x,)
S2 =-- ---
_n-1[ X
P._2(x)P-,(X,,) Pn-.(x)P.-2(X,)
(X, x)2
n-6 2 2 k Pk-l(X)P (x)-Pk(x)P-l(x) k k) _- Pk(x-)Pk-l(x1)
-6 AAk+l\ x x (X, a)2
k=2


(n 1)P_--(x)P'_2(x, ) (n 1)Pn-2(x)Pn-i(x,) (n )P+-(s)P,_2(,)
An- (x XL) An-l(x X,)2 An-1(x x)2

6 n-2 -2 (Pl())P2))
6(
+-Pk- (X) Pk( x,) Pk(X)Pk-(x,))
Sx zx, AkAk+l
k=2





From (2.1.4) and (2.1.5) we have
(3.2.4)

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

(3.2.5)


Pn-2(X ) = XPn-l(z)








(3.2.6)

P,_2(x) 7n + zPn-l().
n--1


Using (3.2.4)-(3.2.6) we combine the first 3 terms in S2 to obtain


(n 1)Pn_-(x)Px2(x,) (n 1)P,_2(x)P.-i,(,,) (n 1)Pf- (x)P,_2(x,)
(n-l(x xy) An-,(Z X2)2 (n-x(Z X,)2
(n- 1)2P,_ (x)Pl) (n- 1)P-1(x,) (7r,(x)
An-1(x x-,) An-(x x,,)2 n 1
(n 1)xPn-l(x)Pn-i(x,)
An-l(x X,)2


(n 1)P,-l(x)Pn-i(x,,) Pn-i((x,)rn,(x)
An-l(x x,) An-1(x ,)2 '


and multiplying through by 4 )4p5(. ) yields


n(n 1)P,~ (x)Pn-,(x,)r (x)(1 x) r(x)(1 x\)
4AXn-(x x,,)n4(n 1)4p,_,(x2,) 4AXn-1(x x,)2n4(n 1)4 P4_ (x)'


where the first term above cancels with the aforementioned term in S1, and we
estimate the second term above using (3.1.9), (3.1.5) and (3.1.11) to obtain


( sin4 t



To complete the first part of the proof, we need only estimate the terms
(3.2.7)

37r (x)(1 x) n"-2 k2( x k(x)P )
2(x x,)n4(n 1)4Pn,_ (xs) AkA k+l










and

(3.2.8)


T2 = 3r (x)(1 ) Pk- )Pk ) Pk )Pk-
2(a X,)2n (n 1)4P5_(_ ) k= k


As Pk(-x) = (--)kPk(x) and xn-,+l = -X,, v = 1,...,n, we may assume that

-1 sin4 tV
We show first that IT21 = 0 (n6 .2 We break this part of the proof into

two cases. We begin with the case -1 < x < 1. Notice

(3.2.9)

|Pk-i(x)Pk(2.)-Pk(x)Pk-1 (X) = I Pk(x)[Pk-(X1 )+Pk(x)]-Pk(X)[Pk-l(X.)+Pck(X)]

< IPk(x,)IlPk-1l() + Pk(x)l + IPk(x)IlPk-1(x,) + Pk (x)l,


so that applying (3.1.3) and (3.1.6) yields

(3.2.10)

Pk-l(x)Pk (x) Pk(x)Pk-1(x,) <


Ssin t
1 +
S--n-- t,
\k sln2 t.


Thus,


n-2 k2
k- 1 (Pk-1(X)PkI(x) Pk(x)Pk-l())
k=2 kc +l


I t" n-2
sin2 L sin2 L_ k
=0 + ---- t k
n4 sini t, n4 sin2 / k=2


(n2 sin2 t,


sin2 t ,
I1
n2 sin t)
+ --- -- .


sin- t,
k sinz t


Since











( 2n4(n 3-1(x)(1 x)
2n(n 1)4P (xx)(-x


S3 9
sin2 t sin2 t,
n4 sin-2 -- sin2 t+
2 2)


we get by using (3.1.11)


sin2 t sin4 t, sin t sin5 t
IT2 = 0 n6 sin2 1 sin2 tt + n6n sin2 t- sin2
2 2 2 2


o sin4 tt,
=0 n6 sin2 t-t,
2


Now we consider the case when 1 < x, < 1. Since -1

. Applying this, (3.2.9), (3.1.5), (3.1.3), (3.1.9) and (3.1.11) to (3.2.8) yields the


estimate


IT1=o(n6 sin 4 t 50
nsin6


sin4 "
Sn6sin2 t
2 )


in this case.


Lastly, we show
( sin4 t,
IT,1= 0 n6sin2 t_ 2

Using (2.1.6), we observe that


(3.2.11)


n-2

k=2


n-2 k2
= E AkAk+l (Pk-l(x)P-2(,) + (2k 1)Pk-(x)Pk-,(x,) Pk(x)Pk_-(x))
k=2


n-2 k2pkl(X)-V) n-2 k2 n-2
= k k -2(X- ) +k+ (2k-1)Pk-l(x)Pk-l(X)-
k=2 kk+1 k=2 +k=2


k2Pk+
AkAk+i


Now, combining the first and third sums above, and applying an Abel transformation


k2
kAk (Pk-1(2)P(X2) Pk(X)P_1(x5))
AkAk+n








with the factors (k+1)2 to the second sum (after reindexing), we have
(3.2.12)

(n 2)2Pn-2(x)P-3(x) n Pk- (x)P2(x) k (k )2
L An-2n-1 A k Ak+l Ak-

n-3
+ (An- 2 k + 1)Pk-()P k(x,)
An-ln-2 k=1
n-4 1)2 )2 k-i
+ ( (k + 1)2 (k + 2)2 ) (2s + 1)P.(x)P.(x,)]
k=2 ik+i k+2 k+2 k+3 s =1


We then apply the Christoffel-Darboux formula (3.2.2) and that


k2 (k 1)2 (2k 1)(5n(n 1) + 3k(k 1))
Ak+l Ak-l Ak+lAk-1


(2k 1)Ak
Ak+lAk-l


to obtain

(3.2.13)

(n 2)2P,_2(x)P_3(x,,) (2k 1)Ak ,P
-An-2n-1 k=3 Ak-AlkAk+li

S(n[ 2)3 PPn-3()Pn-2(X,) Pn-2(X)Pn-3(X,)
An-2An--1 x

(2k + 3)Ak+2 Pl(Pk- )Pk(x,) Pk(x)Pk-l(xv))
k1=2 k+lAk+2k+3 Xv --


U1 + U2.

Using (3.1.3) and (3.1.4), estimating term-by-term we obtain
(3.2.14)

Ul = On2 sin .
n2 sin2 t sin2 t,,





37

We break the estimate of U2 into 2 cases. First suppose -1 < x, I< 1

Applying (3.2.10) to U2, and estimating term-by-term we get


(3.2.15)


IU = O sini t
n sin2


sin 1 I
n2Ix aj[sins2 t,


Now, supposed < x, < 1 (from before we need only consider -1 < x < 0). Then
1- < 2, and we have on using (3.2.9) and (3.1.3)
(3.2.16)
(3.2.16)


U21 = n2 t si t
n2 sins2 tsm22 t)


+0 (n-4


E(2k + 3)k(Pk-(x)Pk(x=) Pk(x)Pk-2(x))
k=2


( 1 Enk=(2k + 3)
=0 .1 1
n2 sins t sin t, n sin2 t sin. ty,


=0 2 1 1
n2 sin2 t sin2 t


Thus, from (3.2.7), (3.2.11)-(3.2.16), and then (3.1.9), (3.1.5) and (3.1.11) we obtain


37r(x)(1 x~)
T1 = 2(x x,)n4(n -1)4P_ (X,)


1( sin? ti
1 i tsm2 t,
0 n2 1 S 3 2 -
(nsh fsin ^ n2si xt\ ssin'


smn2 t
Sn2 -1- sin2 ti


=0( sin4 tv
= n6 sin2 t-t


This completes the proof of the case when It tI > .

We obtain the second estimate in the lemma using the estimates


I7r'(x)7rk(x) 7r,(x)7r' (x)I = 0 (nk~ ,








and

(3.2.17)


v=2,...,n- 1.


Thus, using the above and (3.1.5) and (3.1.9), we have when It t,I < -


sin t sin2 tI
=0
n2


n ()(1 XP )
n4(n 1)4'_P (x,)


and


1(2k kl)Pk (x,) (7r'(x)rk(x) 7rn(x)7r(x))


Hence the proof of the lemma is complete.

Hence the proof of the lemma is complete.


Lemma 3.2.2 We have


n-1 E,(x)l O(n- ).
i (1 (1 2-
1,- ---2 -
Sd^^ '" )


Proof

By using Lemma 3.2.1 and (3.1.11)


(n n-1
S=2 sin t, sin 1^
__ 2


+ n-1
+ n-6 E sin2 t-1t
v=2 2


The two sums above can be found in the paper of Szabados and Varma [43], page


= sin t
=0 -"-


n-1


= O(n-4).[


sin t = O(sin t,),








3.3 Estimate of the Fundamental Polynomials of the Third Kind


Lemma 3.3.1 We have

(3.3.1)


IDI(x)l = O(n-6),


n--1 ID .(x)l
1 -2
v=2 -- X


I= X2
v=2 1


ID.(x) = O(n-6),


( log n



-=o ,
n3 ,


}- = O(n- ).


We have on using (2.1.1), (2.1.8), (2.1.11), (3.1.4) and (3.1.5)


6n3(n -(X)
6n 3(n -- 1)31


and using (3.1.3) and Lemma 3.2.2 we get


-4n-1 P-1(Xk) E(

-4 f( Ek(X
k=2 (Xk 1)


/ n-1
=0 n-2
k=2


) = (n -1 () -Ek )

ck=2 /k

IEk(x) (n-),
(1 )2
kI ) = r )


so that IDj(x)I = O(n-6). Observing that D,(x) = -Di(-x) gives (3.3.1).
We note next that for It t, I< on using (3.1.5), (3.1.9), (3.2.17) and that

le,(x) < 1, we have
(3.3.4)


r(6) (n) 3((x)
6(1 x2)n3(n 1)3P3_(xL,)


X= 0) (( 42 = 0(n S)
6(1 )(1 )n( 1)n
6(1 x)n3(n 1)3 (n )


(3.3.2)


(3.3.3)


Proof


= (1 + x)P'I_(x)7r ( n-)
= O ( o(- )
I 6n (n- 1)<









and further using (3.1.7)-(3.1.8) we obtain
(3.3.5)

(x) e,(x) / (1 2)2nt)(l )n
6(1 x)2n3(n 1)3P_(x,) \ 6(1 -x z)2n3(n 1)3


= O = O(n- ).
S6(1 )(n 1) O(n-2)


Now, applying (2.1.8), (2.1.3), (3.1.5), (3.1.9) and (3.1.11) we have
(3.3.6)

C 7r1w (X),
S 6(1 x )n3(n 1)3P,_1 (x)
It-tvl>


6(1 X2)n4(n- 1)4P_i(x,)(x- x,)


0 ( sint
It-tvl> C v\' ~Vs"


= (n 1)4 sinn It- sin t
It-t 2 2

(n sint (log n
It-tI>E sin n3

Next, using (2.1.8), (2.1.9), (2.1.3) and then (3.1.3), (3.1.9), (3.1.11) and Lemma
3.2.2 we obtain
(3.3.7)
n-~ n-(-
Sn-i) 4
E_ P2-1_P, _2) ZPl(Xk)et(Xk)Ek(x)
J /=2 k=2

n-_1 Pl_ (Xk) kX
v=2 Itk-tk >










SIEk(x)
(1 x2)2 t
Itk-tl>n


(1 -- X)2p4_- (Xk)
(1 -- x)(xk )P1 (x,)


I Ek(x)j
k(1_ x2)2 L
SItk -t- >-


= O nlogn1 E(1 X 2)2


= log n)
=O( n3
0(').


We note that when v = k, the double sum in (3.3.7) is 0 as ef(x,) = 0, v = 2,..., n-1.

Observing (3.3.4), (3.3.6) and (3.3.7), we see that (3.3.2) holds. To show (3.3.3), we

observe (3.3.5), and argue in a manner similar to (3.3.6) and (3.3.7) to obtain


~r6((x) 3p(x)
6(1 x2)2n3(n 1)3P31 )


= O n-4
v=2


1si sn
sin t, sin Ll


= 0 (n-2)


n- 1 n-1
( )P4 P-,2E _, (xk)i,.(xk)Ek(x)
=2 (1 2 (2 ) k=2


( IE(x 1 ) (n-2)
(1 -x2D sin t, sin


This completes the proof of the lemma.

3.4 Estimate of the Fundamental Polynomials C,(x) and C,(x)


Lemma 3.4.1 We have


IC,(x)l = O(n-4), ICn(x)l = (n-4).


ssi tk k
sin )
2


=0 (4
k=2


(n-1

k\=2


=n-O

kc=2









Proof
Using (2.1.3), (2.1.8), (2.1.11) and (3.1.5) we obtain


(1 x)xr(z)() (1 x)~rx(z)
4n(n 1) 4n(n 1)(: 1)27r(1)2



4n3(n 1)3(1 = n


Using (2.1.8), (2.1.11), (3.1.5), Markov's and Bernstein's inequalities


2n2(n 1)2


2nn 1( )2(1)2
2n2(n 1)2(x 1)27r,(1)2


=0 (1 ) ()PP-I(x)rn(x)
S 2n4(n- 1)(x 1)2


((1 + x)2n (1 2)n (n-4),
2n4(n 1)4


and finally, using (2.1.9), (2.1.3), (3.1.3), IPn-i(x)l < 1 and Lemma 3.2.2


n-1
-12 z Pn-(l(xk) k)ex2(1
k=2


+0 (n(n 1)


S n-1l P-I(k) E( )
Sk=2


+0 (n(n


n(n 1)(
2 (zk -


1))Ek(x)


n-1
Xk 1)P2_1(xk) (xk)Ek(x)
k=2


n-1 4
L- 1) 4 )--1
k=2


n-1=2
k=2


P"i(kX k)2Ek(x))










Ek(x) n(n 1) n- 1 1 Ek()
(= 1 k--2k2 2( ) X2)2 Xk
k=2 k=2


= ( Ek(X) =(n-').
_(1- X2)2
k=2 k


Noting that C,(x) = Ci(-x) completes the proof.


3.5 Estimate of the Fundamental Polynomials of the Second Kind


Lemma 3.5.1 We have
(3.5.1)


IB(x) = O(n-2),


(3.5.2)


n-1 B
IB( x)\
(1 ( 2)-
v=2 v I


Proof


We show first |Bi(x)l = O(n-2).
inequality and ef(x) < 1 we obtain


7,(X~) (X) 7 { (x)1(()


= (1 + X)2)(1 -
Sn2(n 1)21-


Applying (2.1.8), (2.1.11), (2.1.1), Bernstein's


S (1- x 2)pl-(x) (x)2(x))
(x 1)n2(n 1)2

)p._l()2) = O(n-2).


Next, applying Markov's inequality (to e (xk)), (2.1.8), (2.1.3), (3.1.3), IPn-l(x) < 1
and Lemma 3.2.2 yield


24 1
24 n-I 7r(Zk)e (Xk)3Ek(X)
n(n 1) k=2


= ( (= X (k) Xk)EkX)
k=2


IB(x)I = O(n-2)



0(log n)









O ( n-21 Pn 1( k)
k=2 (1 xk)2 k )


=0 ( n2 (Ek()2 = O(n-2)
k=2 k );/


Observing Bn(x) = -Bi(-x), we deduce that (3.5.1) holds.
Using (2.1.1), (3.1.9), (3.1.4), (3.2.17) and |l3(x)i < 1, we note that for
It t.\ < -n


nr(x) (x)
(1- x)in(n- 1)P,_ (x,)
S-)Z 2 pn,1
^^/(l^)n P


(1-V)=4n2(n-)

S0(n-1).


Observing (2.3.11), (3.3.2) and (3.1.7)-(3.1.8) we have


n-1
S4n(n 1) I(
v=2 1X


n-1
0 (2 E
v=2


n-1
n 13n(n- 1)x,
E=(1 -X)"(1-- X2)2


(log n
Vn

IE,(x)\
(1--),


= O (n3 |n-1 E_.(x) >= (
n-1(1 X2)2


Now, applying (2.1.3), (2.1.8), (3.1.5), (3.1.9), (3.1.11) and e2(x) < 1 we obtain


S7rn(x)(x)
It- (1 -- x)2n(n 1)Pn-_1(x,)


0 sin tn2(n 1)2P,21(X,)_ -
\t-t ,l> -(n?


n-1
n 21 I D.(x)\\
Sv=21-z 2
1 =22










sin t
(n 1)2X xr


-o n-Y,
=o(n- 2
I -t'i>i


SO log n)
\ n


sin
sin h2 )


and further using (2.1.9) and (3.1.3) yield


n-1 24 n--1
24
Ev2 (1 2) k=2


Ek(x)
(1 ~)2


Itk-tLl>-


7rn(k k)
Ek(2)


P'-L(Xk)(1 XD)2
(1- x,)Llxk X3P, ,)
k2 22_,(X,)
k 3 4-1i


( IEk(x)
k tk-t\l>


n -1
0 (n3E
\ k=2


IEk(x)
( k )


1i )
sin3
2


= O(n-).


This completes the proof of the lemma.


3.6 Estimate of the Fundamental Polynomials of the First Kind


Lemma 3.6.1 We have


SIA~(x)l = 0(1).
v=1


Proof

We show first that IAi(x)l = 0(1). It suffices to note that on using Bernstein's


inequality, it (x)I < 1 and Lemma 3.2.2


IEk ) = (1),
(1 Zk2/'


n-1 n-1
S(Xk)' E,(x) = O n4
k=2 k=2


El>
G*I>{


= It


(n-1
k=2
kf=2


S(n-
k=2





46



and observing A,(x) = A,(-x), we deduce IA,(x)| = 0(1). Now, as fe(x) < e~(x),

we have


n-1
Ze4 (x) <
v/=2


n-1
3 ,e(x)) <1
v=2


and observing (3.3.2), (3.1.7)-(3.1.8) and Lemma 3.2.2 we have


( 0 n-12 IDV(x)I


S log n)
n


IE,(x)
(1 zx)2


n-1 E \x1)3


n-1
S1 E,- (x)
v=2 (1- z)=


) =0(1).


It remains only to estimate

n-1 n-1
S(- 24 Z" (Xk)4Ek(x)
v=2 v=2


and applying (2.1.9), (2.1.3), (3.1.3), (3.1.9) and (3.1.11) yields


(n- (IEk(X) I
\E(1- x2)2 c
k=2 Itk-t| l>-

n- ( Ek(
Ik(1 X tlI)2
\k=2 Ith-t l:


(1- x2)2p,(xk)
(Xk X )4P4_ 1(x)



-c k x,4


4n(n- 1)x,
v=12 -
v=2 "


S24n(n 1)
v=2 5(1 x2)2
v.=2V


Sn-1
=0 n E
v=2


n2 n + 3 Ex(2) )
\ 1 4


= 0 (n4










= (I x ) sin2 tk sin2 t,



\k=2 2 I tk-tI> 2
\k=2 tk- s> n 2


=0 _k(x) = 0O(1).
Sk= (1 k)


This completes the proof of the lemma. O

3.7 Proof of the Convergence Theorem

Denote by [x] the largest integer less than or equal to x. This expression is

known as the greatest integer in x. Let f C[-1, 1], m = and consider

polynomials pm(x) of degree at most m such that

(3.7.1)
1
IIf/)(z) p )(2X)I = O(m()w4(f, -
m

and

(3.7.2)

IIP$)(x)II = O(m()wj(f, -), 1 < j 4
m

The above polynomials exist by a paper of H. Gonska [16], page 165.

Since 'modified' (0, 1,3,4) interpolation is uniquely determined, we have
n n
pn(x)-Rn(Pm, x) = Ep',(x,)B,(x)+p (1)Ci(x)+p(-l)Cn(x)+E .(x.)D,(x)
v=1 v=1
n-1
-+- Ep)(x.)E,(x).
v=2


Thus, using (3.7.2) and Lemmas 3.5.1, 3.4.1, 3.3.1 and 3.2.2 we have










n n
IPm(x)-R(pm, x)l = Zp'(x,)B,(x)+p (1)Ci(x)+p"(-l)Cn(x)+L p'"(x,)D,(x)
v=l 1L=l
n-i1
+ E PM)(xv,)E(Zx)
v=2
(3.7.3)

n1
= o(m)wi(f, 1) IB.(x) + O(m')w2(f, -)(ICi(x)l + IC(x)I)
v=1
n n-1
+o(m3)w3(f, lD.() + O(m )w4(f, 1) |.(2)l.
V-=1 v=2


Now, from (3.7.1), (3.7.3) and Lemma 3.6.1 we get



Ilf Rn(f)II < Il PmII + 1pm Rn(pm)I + IIRn(pm f)ll


5 Ilf Pmll + Ilpm Rn(pm)II + -ip, -


+ 0w (f, log n
+o^(/,^n


fll II E IA,(x) II
uV=1

+ 0 (W4f, )
n/. -


(f logn
-


This proves the theorem.


W4 log n


= 0 (wj














CHAPTER 4
ERDOS-TYPE INEQUALITIES

4.1 Main Results


Let us denote by S, the set of all polynomials whose degree is n and whose

zeros are all real and lie outside (-1, 1), and denote by L, the set of all polynomials

of the form

(4.1.1)
n
Pn(x) = akqnk(x), ak >O (k =0,1,...,n)
k=O
where qnk(x) = (1 + x)n-k(l X)k

Here we present two theorems concerned with finding a uniform upper bound

for the expression
S1 w(x)(Pn(x))'dx
1, w(x)(P,(x))Pdz


where w(x) = (1 xa2)a,a > -1, when p = 2, and w(x) = (1 x2)3, when p = 4,

and where the polynomials P, are restricted to the Lorentz class Ln of polynomials.

It is known [22] that if P, E Sn, then P, E Ln or -Pn E Ln. Thus, Theorem

4.1 is an extension of the classical theorem of P. Erd6s [10] for Pn in Sn, as well as

the results of P. ErdSs and A.K. Varma [11] and the Theorem 3.4 in Milovanovi6 and

Petkovi6 [28], in the L2 norm with the ultraspherical weight w(x) = (1-x 2), a > -1.

In Theorem 4.2 we present the first sharp extension of the inequality of Erdos in a

weighted L4 norm. Note that Theorems 4.1 and 4.2 provide the polynomials which

attain the given upper bounds.







We shall later see that for each n there exists a unique positive solution to the
equation

24 + (8n-5)a3 +(12n2 -17n+4)a2 +(8n3O- 20n2+lln- 1)a-2n(2n2 -5n+4) 0.

Denote this solution by an.


Theorem 4.1 Let Pn E L,, n > 2 and a > -1 real. Then we have for a > an

(1 x2)a(P'(x))2dx < 2(2n + 2 n + (1 x2)'(P())d
12(2n + a)(2n + a 1)

with equality for Pn(x) = c(1 z)n. For -1 < a < an the inequality becomes

j(1 )P,, ))2d

<(2n + 2a + 1)(n + a)[a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))]
2(a + 1)(a + 2)(2n + a 2)(2n + a 3)

x (1 x2)(P 2dx
1-
with equality for P,(x) = c(1 x)n-l(1 t x).


Theorem 4.2 Let Pn E Ln. Then we have


f' 21 I < n3(4n + 7)(4n + 6)(4n + 5)(4n + 4)
n-( 64(4n + 3)(4n + 2)(4n + 1)

with equality if and only if P,(x) = c(1 x)".

4.2 Some Lemmas

Lemma 4.2.1 Let P, E L, and a > 0 real. Then we have
(4.2.1)

(-x))2 dx <(2n + 2a + 1)(n+ a) (1
-1 2a(2n + a -1


S(1- -x2)3(P,(x))4dx
_z ^^)4


x ), P 2))2dx








with equality if and only if P,(x) = c(1 x)". In the case P,, L,. and P,(1) = 0,
the inequality becomes
(4.2.2) (1 2a- <(2n + 2a + 1)(n + a) (I_ 1 2)1(P(X))22dx
(4.2.2) (1 x2)'-i(P,(x))2dx < 2 )2+ (l 2-)a(Px))d
1-1 2(a + 2)(2n + a 2) -1
with equality if and only if P,(x) = c(1 x)n-'(1 F x).
Proof

We write Q2n(x) =(Pn())2. Then Q2n E L2n and we have from (4.1.1)
2n
Q2(x) = ak(1+ )2n-k(l .
k=O
Thus,
(4.2.3)

f~-,(1 x2)_-l(P,(x))2dx ko ak fl(1 + )2n+a-l-k(1 _- )k+-ldx
_1(1 x2)a(P,(x))2dz o ak (1 + x)2"+c-k(l x)k+aod
and we use the known formula
(4.2.4)
IF(p + 1)F(q + 1)2p++
-1 F(p + q + 2)
to obtain


fj(1 + x)2n+-l-k(l )k+a-ldx r(2n+a-)r(k+a)22"+2a-
__-l_____ ____________ ;_ Pr(2n+2a)
f(1 + x)2n+.-k(1 x)k+adx r(2n+a+l-k)r(k+a+1)22n+2a+
1 rF(2n+2a+2)
S(2n + 2a + 1)(2n + 2a) (2n + 2a + 1)(n + a)
4(k + a)(2n + a k) 2a(2n + a)


for k = 0,1,2,..., 2n, where equality holds iff k = 0 or k = 2n. Applying the above
to (4.2.3) yields (4.2.1). If P,(+1) = 0, then we note that k runs only from 2 to
2n 2 above. Employing this observation in (4.2.3) proves (4.2.2). This completes
the proof of the lemma. O








Lemma 4.2.2 Let P, E L, and a > 1 real. Then we have
(4.2.5)

(1 x2, p dx < an (1 X2),-(Pn(x))2d
J-1 -2n + a -1 -
with equality if and only if P,(x) = c(1 4 x)". In the case that P, E Ln and

P,(1) = 0, we have for a > -1 real the inequality
(4.2.6)

(1i 2)( ))d < a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))
(a + 1)(2n + a 3)


x (1 -x)-(P,(x))dx

with equality if and only if Pn(x) = c(1 z)n-l(1 T x).


Proof
From (4.1.1) we have


(P(x))2 = E
j=0


n
E akajqnk(x)qnj(x)
k=0


so that we may write


- x2)a-'(P(x))2dx


n n=
= E aka=
j=0 k=0


= akaj ( + )2+a-1-k-( x)k+j+-1 dx.
j=o k=O 1


On using (4.2.4) and writing = k + j we obtain

(4.2.7)

j(1 X2)l(P(X))2d" akaFj(2n + a )r( + a)22n+2a-'
1(1 '( )dx = = k= F(2n + 2a)


(1
1


2 (1 x qk()qnj()dx
1-









Next we show that for a > 1



(1 x2)(P(x))2dx = E aka (1 X)qk(X)q(x)dx
-1 j=0 k=0 -1
(4.2.8)
< an 2 akajr(2n + a )r(y + a)22n+20-1
2n + a 1 F(2n + 2a)
j=O k=O

As qnk(x) = (1 + )-k)(1 ) we have

qnk(x) = (n k)(1 + x)n-k- -)k k(1 )n-k(l )k-1

and using (4.2.4) yields
(4.2.9)
[1 22n+2-1 r
SIk 2n d+2cr- [(n-k)(n-j)r(2n+a-1--)F(t+a+1)
Ikj 1--1 )'qnk(x)qnj(x)dx = r(2n + 2a) -

+kjr(2n + a + 1 )r( + a 1) (nt 2kj)r(2n + a )ry( + a)]


r(2n + a )r(e + a)22n+2a-1
r(2n + 2a)

2n+a-1- +a(2n+-2k


We denote the portion in brackets by pkj and simplify the expression as follows,
denoting e = k + j and later using that 4kj = 2 (k j)2. We have


(n2 ni + kj)( + a) kj(2n + a i)
2kj + nI + 2kj
-k= 2n+a-l-1- +( -1


an2 + n2t n(2n + 2a 1) kj(2n + 2a 1)(2n + 2a 2)
2n+a-1-- 2n+a-l-1 + (2n+a-- )( + a 1)









an2 [2n2 + (2n e)(2a2 3a + 1 n)]
2n+a-1-e 2(2n+a- 1-)( +a-1)


(k j)2(2n + 2a l)(n + a- 1)
2(2n +a- 1 -e)(e+a- 1)


2a(a- 1)n2 + (2n- )(n -(a- 1)(2a- 1))- (k- j)(2n + 2a- 1)(n + a- 1)
2(2n + a- 1 )( + a 1)

We show that Iki > 0 for k,j = 0,1,.., n, (a > 1) and apply this
to (4.2.9) to obtain (4.2.8). We have


an2 an2
2n+ a -- 1 k = 2n + a- 1


2a(a 1)n2 + (2n )(n (a 1)(2a 1)) (k j)2(2n + 2a 1)(n + a 1)
2(2n+a- 1 -)(+ a- 1)



> e(2n )[2(a 1)n2 + (4a 3)(a 1)n + (a 1)2(2a 1)] > 0
2(2n +a-1)(2n+a- 1 )(+ a-1) 1

with equality iff f = 0 or t = 2n.

Observing (4.2.8) and (4.2.7) we have (4.2.5). Lastly, for the case P,(l) = 0

(k,j = 1,2,...,n 1) and -1 < a < 1 we have


a(a l)n2 + 2(n l)(n (a 1)(2a 1))
(a+ 1)(2n+a-3) -- kj

(2n 2(n 1)a a2)[f(2n ) 4(n 1)]
(a+ 1)(2n + a- 3)( + a 1)(2n + a 1 )
(k -j)2(2n +2a -1)(n+ a- 1) 0
(f+ a-1)(2n + a-1 -)










as 2n 2(n 1)a a2 > 1 for -1 < a < 1, and (2n t) 4(n 1) > 0, with
equality iff k = j = 1 or k = j = n 1. This yields


(1 x2)"(P'(x))2dx


a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))
(a + 1)(2n + a 3)
S akajr(2n + a )r( + a)22n+2a-1
j=0 k=0 r(2n + 2a)


Combining the above with (4.2.7) yields (4.2.6), completing the proof of Lemma 4.2.2.

O


4.3 Proofs of Theorems


Proof of Theorem 4.1
Let Pn E L,. The case a > 1 follows from (4.2.5) and (4.2.1). Now, let n > 2 and
-1
P,(x) = ao(1 + x)n + Q,(x) + an(1 x)"

where Q, E L, and Qn(1) = 0. Then


(4.3.1)


f(1 x2)(P(x))2dx = n2a2
1-i


J (1 x2)r(1 + x)2n-2dx
1-


+n2a 2 (1-x2 )2n-2dx+ ( 1 x )a(Qn(x))2dx
21 -1

-2n2aoan (1 x2)n+-ldx + 2nao (1 x2)(1 + x) Q'Q(x)dx
-1 J-1
-2nan (1 x2)a(l x)b-'Q'(x)dx.


We show the last 2 integrals are nonpositive. Integrating by parts we obtain











II = 2nao (1 2 )'(1 + )-'Q'(x)dx = 2nao (1 x)a(l + x)"+-lQ',(x)dx
J-1 -1


= -2n(n + a 1)ao (1 + x)"+(-2(1 -_ X)Qn(x)da
J-1
1
+2naao 1 (1 + X)"+-1(1 x)a-lQ.(x)dx.
J-1


Thus, we have II < 0 for -1 < a < 0, and if for 0 < a < 1 we show

(4.3.2)
S1,(1 + x)n+a-l(1 )a-lQ.(x)dx 2n + a 2
+(1+ Z)" ( -- z)Q(x)dx 2a


then we will have

fl
I, 5 -2n(n + a 1)ao (1 + x)n+"-2(1 x)Qn(x)dz


+(2n + a 2)nao (1 + X)n+-2(1 z)oQn(x)dX

r1
-aaon I (1 X)a(l + x)n+o-2Qn(x)dx < 0,
J-1


as desired. We show now (4.3.2). It suffices to consider


S(1 + x)n+-lx(1 x)'-'qnk(x)dx
f,(1 + x)n+O-2(1 x)aqnk(x)dx

f 1(1 + )2n+a-1-k(l X)k+a-ldx 2n + a 1 k
f_1(1 + X)2n"+"-2-k(1 X)k+fa k + a

2n+a-2
< + (k =l 12, ...n 1)
+l









2n +a -2
2a


(0 < a < 1)


showing (4.3.2). In the same manner we obtain



2 = -2na (I- x2)a(1- )"-'Qx1(x)dx = -2na, (1-x)n+(-'(1+Xz)Q',()dx
-1 J-1


= -2n(n + a 1)a,, (1 + x)*(1 x)n+"-2Qn(x)dx
J-1
1
+2naan (1 + x)-'(1 X)"n+-'Q.(x)ds,
J-1


showing that I2 < 0 for -1 < a < 1. So from the above, (4.3.1) and (4.2.4) we obtain



(I1 _r 2)((2n + a 1)r(a + 1)22n+2a-'
/(1 nn2-) F(2n + 2a)

+ j(I z)(Q ))2d
J-1


and noting that


1~ 2)(P r(2n + a + 1)r(a + 1)22n"+c+1

0 n rF(2n + 2a + 2)

+ (1 X2)(Qn(x))2dx
J1-1


yields


(1 x2)Y(p'(x))2dx
f_(1 x2)(P,(x))2dx








(4.3.3)

n2 ) F(2n+a- )r(a+1)22n+2a-1 ))2
n2(ao + a )(2na+l +22 + ,(1 z2)a(Q,(x))2dx

Now consider + a)(Q )d
Now consider


f(1 x12)(qo(X))2dx
_, (1 x2)l(qno(X))2dx


f,(1 x2)(ql(x))2dx
,1(1 x2)a(qnl(x))2dx


(n 1)(2n + 2a + 1)(n + a)f,(a)
(a + 1)(a + 2)(2n + a)(2n + a 1)(2n + a 2)(2n + a 3)'


where

fn(a) = 2a4+(8n-5)a3+(12n2-17n+4)a2 +(8n3-20n2+lln-1)a-2n(2n2-5n+4).

As the denominator of the above ratio does not change sign, we need only look at
fn(a). It not difficult to check that f,(a) is increasing on (-1, oo), and that it has
precisely one positive zero, which lies inside (0, 1). We call this zero an. From above,
we see that for -1 < a <- a,


n2(2n + 2a + 1)(n + a) <(1 x2)*(q~o(x))2dx fl(1 x2)a(q())2dx
2(2n + a)(2n + a 1) f 1(1 x2)a(qo(x))2dx f(1 x2)c(qn,(x))2dx


(2n + 2a + 1)(n + a)[a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))]
2(a + 1)(a + 2)(2n + a 2)(2n + a 3)


and for a > a, the above inequality is reversed. Employing this observation in (4.3.3)
completes the proof of Theorem 1. O








For the Proof of Theorem 4.2 we shall use the inequality for P. E Ln and -1 < x < 1

(4.3.4)


(1 x2)((PF(x))2 P,(x)P"'(x)) < n(P,(x))2 2xP,(x)P,(x).


This inequality is found in the paper of MilovanoviC and PetkoviC [28], page 284.

Proof of Theorem 4.2
Multiplying (4.3.4) through by (1 x2)2(P,(x))2 and (1 x2)(Pn(x))2 we

obtain the inequalities


(1 x2)3I(P(x))2((P,(x))2 Pn(x)P'"(x))


(4.3.5)


< n(1 x2)2(P,(x))2(P, ())2 22x(1 x2)2P()(P'(x)3


and

(4.3.6)


(1-x2)2 (Pn())2((P:(x))2-Pn(n)P2())n < n(l- 2)(Pn(x))4-2x(1-x2)(P.(x))3P,(xt).

Denote

= (1 x2)3(P(x))4dx.
-1
Integrating by parts yields



1 1
I= -3 j1 (1 X2)3(p(x ))2P.(X)P,'(X)dx + 6 j x( X2)2(P, (x))3P,(x)dx,



and adding 311 to both sides we obtain



411 = 3 (1- (()2((,(x))2-P"(x)P(x))dx+6 (1-x2)2(p(x))3Pn(x)dx.
1 1









Applying (4.3.5) to the above yields the inequality
(4.3.7)


4 (1 x)((x))dx < 3n (1 P(x))((dx.
1 1


Now, for any polynomial P,(x) we have

d
4(1 x2)2(Pn(x))2(P()2 )2- [(1 x2)2(p(s))3p(x)]

= (1 x2)2(P,(x))2(P(x)2 + 4x(1 x2)(P,(x))3P,'() (1 x2)2(P,P("().


Thus, integrating both sides of the above from -1 to 1, and then applying (4.3.6) we
obtain


4 f(1 x2)2(P,())2(P,(x))2dx =
JI


(1 x2)2(P,(x))2((P'(x))2 Pn(x)P.(x))dx
-1


+4 x(1 x2)(P,(x))3pn(x)dx
Il 1l


11 1



Integrating by parts the last term above yields the inequality


4 (1 x2)2((Pn ))2(p :())2dx < (n -)
-1 2 f,


(
(1 x2)(P(x))4dx + (Pn(x))4 dx.
--1


Applying the above to (4.3.7) we get









( x)(P,(x))4dx < 3n(2n- 3) 1 (1 2)(P,(x))4dz + 6 f(P())dx.


Now, replacing n by 2n in (4.2.1) with a = 1, and then a = 2 and a = 3 yields


l(1z-2)3(P,(x))4dx <
J 1-


3n(2n 3)
32


3n(4n + 3)(4n + 2)) /1
64(4n + 1)


=2n (1 x)(P,(x))4dx
2(4n + 1) 1


n3(4n + 7)(4n + 6)(4n + 5)(4n + 4) (1 )(P.))'d
- 64(4n + 3)(4n + 2)(4n + 1) J_


completing the proof of Theorem 4.2.


(1-x')(Pn( x))dx












CHAPTER 5
TURAN-TYPE INEQUALITIES
5.1 Main Results

Let Hn denote the set of all polynomials of degree n whose zeros are all real
and lie inside [-1, 1]. Inspired by the inequality of Turin [45], we present three
theorems concerned with finding a uniform lower bound for the expression
1, w(x)(P'(x))Pdx
w (x)(P ,(x))Pdx
for P, E Hn. We provide sharp inequalities for the special cases p = 2 and w(x) =
(1 x2), a > 1 (a > -1 if P,(l) = 0), and p = 4, w(x) = (1 x2)3. Then
we present an asymptotically sharp result for w(x) 1 and p even. Theorem 5.1
generalizes some previous results of A. K. Varma [51],[50], and Theorem 5.2 extends
these in a weighted L4 norm.


Theorem 5.1 Let P, E H, ,n > 2 and a > 1 real. Then we have (n = 2m)

(1 n2(2n + 22(p)) + 1)2 J (1 -x2)a(P,(x))2dx
1 4(n + a 1)(n + a) -1
with equality if and only if P,(x) = c(1 x2)m. If P,(1) = P,(-1) = 0, then the
above remains valid for a > -1.



Theorem 5.2 Let P, E H,. Then we have (n = 2m)
1 x p 3n3(4n + 7)(4n + 5) > (1 )(()) d
1( x)3(P(x))4dx 4(4n + 6)(4n + 4)(4n + 2) (1-
with equality if and only if P,(x) = c(1 x2)m








Corollary 5.2.1 Let Pn E Hn. Then we have


(5.1.1)


(1 )(P(x ))dx> 8) (1 x2)(P,())4d
>1 -8(2n + 1) _1


and
(5.1.2)


I 3n3(4n + 5) (P())
(1 x2)2(P())4d > 32(n + 1)(2n + 1) _


where these results are the best in the sense that there exists a polynomial P*(x) of
degree n having all zeros inside [-1,1] (P,(x) = (1 x2)m,n = 2m) and for which


and


f-,(1 x')(P,*'())4dx

f(1 x2)2(p,'(x))4dx
Sf,(1 xz)2(P*(x))4dx
f(1 I2 2P.*(X))4dz


3n3(4n + 3)(4n + 1)
64(2n + 1)(2n 1)(n 1)

3n3(4n + 5)(4n + 3)
64(n + 1)(2n + 1)(2n- 1)'


Theorem 5.3 Let Pn E Hn and p > 2 even. Then we have


((p l)n (p 2))((p 3)n (p 4)) (5n 4)(3n 2)n(pn + 1)
p (pn 2)


where this inequality is sharp in the sense that for the polynomial P*(x) = (1 x2)"
(n = 2m) we have


1 IIP,*'I (p-1)(p-3)...5-3
lim L
n-oo n2 IIPnl p2


(Pn(x))Pdz
J-


(P'(x))Pdx >





64


5.2 Some Identities

We shall need the following known identities.

(5.2.1)
1
P (x) = P (x)Z 1
X Xk
k=l
(5.2.2)

(P()) P,(x)P,'(x) = (P,(x))2 1 -(-
k=1 (Z Xk)
(5.2.3)
1 2 2x 1 z
+ -- = 1+
(x xk)2 x- Xk ( Xk)2


Identity 5.2.1 Let a > 0 and P,(x) be any algebraic polynomial of degree n with

x1, X2,... xn as its real zeros. Then we have

2 (1 x2)a+l(P(x))2dx = (n (2a + 1)a) (1 X2),(p))2x
1 1

+2a2 (1 x I-P())dx
1 -1

+ (1 2)(P_,(x))2 2 d.
-(1 k=1 k)2
1 k=l
Moreover, if P,(x) vanishes at x = 1 and x = -1, then the above is also valid for

-2 < a < 0.

Proof

Integrating by parts, we have for a > 0

(1- 2)a+l p ()2dx


=- f- P,(x) [(1 x2)a+lPP,(x) 2(a + 1)(1 x2axP'(x)1 dx.
1-1


Therefore, we obtain







1 1
2 j(1 x2 (p x)2dx = [(P,(x))2 P(X)P:,(x)] (1 x2)+dx


+2(1 + a) (1 x2)a xP(X)P,(x)dx.


Now, on using (5.2.1)-(5.2.3) we have

2 (1 x2)+l(p,(x))2d = (1 2)(P_(x))2 E l d
1 k=1 k)

+ 2(X2) (Pn()x dx-a j(P,(x))2{(1 -2) -2ax2(1 a-d
-k=1 k 1

1- -
S (1 2)(Pn,())2 dx + (n a) (1 X2)"(P,(x))2dx
J- 1 (X k)2 -

-2a2 a2 ( 1)(1 x2)-P ))2d


( 1 x) d2
(1 X2)-(pn(X)) (x xk)2 d

1 k=1 k)

+(n a(2a + 1)) (1 x)=(P,+(x)) dx
--1

+2a' (1 -2 )a-X(P(x)) 2dX.


This proves the identity for a > 0. For the case Pn(l) = 0 and -2 < a < 0, the
above proof remains valid. Thus, Identity 5.2.1 is established. O



Identity 5.2.2 Let Pn,() be any algebraic polynomial of degree n with n real zeros
x1,x2... x,, and let a > 0. Then we have

( X2)-(pn(X))2dx (2n + 2 + 1) pn dx
1 (2n + 2a) _







1 1 1- x
Sdx
-(1 ')(Pe(x))' 1 k dz
2(n + a)2Jl 2 ( Xk)2

+ ( 1 1 (1 x2)a-1[(1 x2)P,(x) + nxPn(x)]2dx.
(n + a)2 I


Moreover, if P,(1) = 0, P,(-1) = 0, then the above is valid for -2 < a < 0 as well.
Proof

First we note that for a > 0




+n2 j( x2)-1'x2(P(x))2dx + 2n (1 ')oxP.(x)P;(x)dx
1 1



ji [- x2 P,)dx + n2 f x2(-xXP))2dx

1- 1 i
-n J(P+(X))2[(1 X2)d 2X2a(1 X2)al]dx



j x2)+l(P()x))dx + (n2 + 2na) x(1 X-))2dx

-n J (P,(x))2(1 x2)dx
1 1




= (1a + )a) (P '( p(X))dd (+2n ) 1 (1 Z2)a-(Px (X))2dz
21 1

+(n 2+2n) 2 (P(x))( 1- 2)-'dx



= ( 1 ) ( )) (P2 +2 z(1 )2 a-1 r( 2dx
2 11 nPn + -11 ))2







1 1 1-2
+- (1 (P(x) 2 dx
2 k=1 (x- Xk)

-(n2 + 2na + n) (1 x)(P,())2dx
1J-

+(n2 + 2na) (P,(x))2(1 x2),-1dx



1J-1


(1 P ) 1-(Pi dx-
+ jni-(2+ 1 (n + 2na + n))P (1 X2,(P. jdx

+ (1 )(P())2 k 2dx ,
--1 k=l


which is equivalent to the stated identity. We note that if -2 < a, then the identity
is still valid provided P,(1) = 0, Pn(-l) = 0. This proves the Identity 5.2.2. O



Identity 5.2.3 Let Pn be any algebraic polynomial of degree n with X1, X2,.. *n its
real zeros. Then we have



(1 X)(P())4dz = 3n (1 x2)2(pn X))2(p'(X))2ds
f(l x p(x))dx = f dx
-1 4 -1
3 r 1- 2 .
+- (1 X2)2(pn)2())2 E ( Xk d.
4 k=1 X -k)


Proof
Denote
J= (1-2)3(p 3 ))4dX.
J-1


Integrating by parts yields










I = -3 (1 x2)3(:(x))2P,(x)P,(x)dx + 6 (1 x)(()) (x)dx



and adding 31I to both sides we obtain


41I = 3 (1 x2)3(P.(x))2 [(P,(x))2 P,(x)P'(x)] dx

+6 x(1 x- xy)i(Pn(x3P(x)dx.
Applying (5.2.1(5.2.3 yields the identity.


Applying (5.2.1)-(5.2.3) yields the identity.


Identity 5.2.4. Let Pn be a polynomial of degree n with real zeros x1, x2,... ,X.
Then for p > 2 even, we have

(1 1
(p r 2) 1 (P )-(P )dx + (r + 2) 1 (1 (P ))))'
J-i J-i


= ((p-r- 1)n (p-r-2))


Jl(P,(X))p-r-2(P(X))r+2dx
J-11


+(p r -1) ))+2 x k) 2d
k-1 =1 x-

+(p- r 2) (P,(X))p-"-2(Pn(x))'(xP,() Pn(x))2dx,
J-1
r = 0,2,...,p- 4.



Proof
Denote
I, = (1 x2)(P())P-'(P(X))rdx.
11-1


Integrating by parts we get










Ir += 1 r (1 X')(P,(x'-))(-r-'P x()P )P."(dx
r+1 j1

r + 1 _-


and adding p--lIr to both sides yields


P I =P-r-l1
rI1 r=_ (1 x2)(P'(x))p -'2(P ,())((P'(x))2 P,(x)P'"(x))dx
r +1 r+1 J-X

P rS 1 j(P^(X))P-'-r(P,(X))r+1dX
2 -r-1 1
+2 (P,'(x))-r-l(pn(x))r+ldzx
r + -1 1-

+( 2( -r- )) x( P(x))p-r-l(Pn(x))r+ldx.
+ r+l r+l -1


We now multiply both sides by r + 1 and apply (5.2.1)-(5.2.3) to obtain


pIr = (p r- )n (P'(x))P--2(P, )r+2d
1-
+(p r 1) f(P(x))r-(P(x))r+2 E 2d
-1 k= (x xk)2dx

-2(p r 1) x(P.,(x))P-r-l(P,(x))r+Idx

-(p 2) j (P(x))P- -(P(X))(XZP'(X)) P d(x))2dx
J-1

+(p r 2) (P,'()),r-2(P,(X))r(XP,'() P.(X))2dx
J-1


= ((p r 1)n (p r 2)) (P'(x))P-(P(x))r+2d
+1 (Xn)r+2 dx
+(p r 1) (Pj'(x))p-S-2(P1(x)) + A d
1 k=1 Xk)
k=1








+(p r 2) (P,(X))p- -2(P,()) (XP,(X) P.(x))2dx
-1
-(p r 2) 2 x 2 X))p-Pn )rdx,
1-1


and taking the last term above to the left hand side completes the proof of the
identity. O

5.3 Proofs of Theorems

Proof of Theorem 5.1
On using Identities 5.2.1 and 5.2.2, we obtain



J-1 -1

1 12
+22 [(2n + 2a)c +1 ) (1 _(-2(pr(l))2) J
(2n + a2) )
-1 -Xk2 dx




-/(1 z) (P,())2 k 2( dz
2(n +a)2 -1 k= k)







+ ) (1 )-' [(1 x2)P(x) + nxP,()]2dx

+ (n- + 2n) -
[ + (1 _2- )(PX))2 k 2d x

na (2n + 2a + 1)


=+ (1 X2 ) .(x2)) (P(x d)) ddx


(n + a)2 -1


n2 1 n2 + 2na 1 1 x)2
= 1 (1 X2),(Pn(x))2dx + (n + j)2 -I_ 2 E k dx
n + a 1 k=1 k)







202 F1
(n+(1 )"-'[(l x2)P(x) + nxP.(x)]2dx
(n + a)2 -

n2 (2n + 2a + 3) 1 (1- _2)o+'(P.(X))2d
n+a 1(2n + 2a+2) 1
1 1 1-2 X
(1 $2) +l(p1n())2 k (5 dz)2
2(n + a + 1)2 -1 )k= )2
k= 1

+ 1 (1 x2)[(1- x2)P,(x) + nxP,(x)]2dx]
(n + a + 1)2 -
n2 + 2na 1 1 -
+ 2 (1 x2)a(P(x))2 E 1 k )2
+(n + )2 -1 k= ( k
292 f 2
+-- 1 (1 x [(1 x2)P(x) + nxPn(x)]2dx.
(n + a)2 -


Note that









Therefore, we have


J- ) (n + a)(2n + 2a + 2) J-1


proving the theorem. O



Proof of Theorem 5.2
Let P,(x) be any algebraic polynomial of degree n with xl, 2,..* as its real zeros.
Then we have

4(1 x2)2( (x)2P)2)(n 2 ( [(1 x2)(P,(x))3P,(x)]
fl -(n +a)(2 + 2 + 2







= (1 x2)2(P,(x))2(P,:(x))2 + 4x(1 x2)(P,(x))3P(x) (1 x2(P,())"P().

We integrate both sides from -1 to 1, and make use of (5.2.1)-(5.2.3). We then
obtain
(5.3.1)

4 (1 x2(Pd(x ))2 = (1 )2(P(x))2 ))2 P.(x)P,(x))dx

+4 x(l x2)(PF(x))3P,(x)dx
1-1


(1
-/'1


- x2)(Px))4 1 -2 + 2x(x xk)dx + 21 (1 x2)(P,(x))3~(x)dx
k=1 ( k -)2 J-


= f(1 x)(P(x))4 k dx + n (1 x2)(P,(x))4dx
J-l k=L J-l

1 /1(1 3x2)(p,(x))4dx
2 -1


1n 1 _X2 3)1 1
=- l(-x2)(P(nx))4 k dx+(n-) I (1-x2)(Pn(x))4dx+J (P,(x))4dx.
-1 k=l ) -1 J-


From Identity 5.2.3 we have


S(1 x2)3(P,(x))4d n (1 2)2 )) 2 p )) 2
J-I 4 -
+ (1 x2)2(P(x))2(P(x))2 (X d
4 -1 k=l k)

Now, using (5.3.1) we obtain
(5.3.2)


1(1 x2)3(p(x))4dx = n [ (
fl 16


-3) x( 'x)(Pn(x))dx + f(Pn())4dx
2/ 11J-








+ (1 x2 (P )4 k 2
1 k=1 k)


+3 (1 x2)2(p,())2(P.(x))2
4 -1 k=l


1-zx-
x -- dx.
(X Xk)2


Also, from Identity 5.2.2 with a = 1 and n replaced by 2n, and on noting that in
this case


1- 2
(x Xk)2


we have


n
?1
=2E
k=1


1- 2
(x -Xk)2


(P(x))d > (4n + 3) (1 2)(P(x))4dx
1 1-x d
-(1 x2)(Pn(x))4 Xk dx.
(2n + 1)2 1 k k


Therefore, on using (5.3.2) and the above we have


(5.3.3)


J(1 x23(Pxdx >


3 (1 2)(P
2) (x))4 dx
2 1


4n+ 3 fld
+2(2 ) 1 (1 x2)(Pn(x))4dx
2T(2n + 1)_I


1 -n, 1 -X2 ]
j (1 x2)(Pn(x))4 k 2kdx
1 k=1 x k)2


3 .1 (1- x )dx
+- (1- x )2P (x))IJx 2 (x- xk )2
-1 k=1


[2n2 (1 2)(pn(n + 1) J 1-))
2n + 1 --(2n 1)2 1 k=1


3 1 3 22p(p\2p'nX )2) 1- xk dx
4 (1- 2)2 Xk)2d.
4 1 k=1


3n
16


1(2 )
(2n + 1)2


1 X2
k)i~











Again, using Identity 5.2.2 with n replaced by 2n and a = 2 we have

(5.3.4)
(4n + 5) / (l x2)2(P,(x))4dx
1(1 x)(P.(x))dx > (n + ) (1 dx
1(4n + 4) -1
(1 2j(i x2)2(Pn(x))4 1(- k)dx
)2 ( ( 2Pk )2 dx,
(2n + 2)2 -1 k=1
and similarly with a = 3


(5.3.5)


l (P())dx (4n+ (1 2)(P(x))4dx
1- 1 u -


1 x 21-
1 (1 x 2)3(pn(x))4 1 k 2dx
(2n + 3)2 1 k=1( X k)d


Therefore, we obtain


(1 2)(Pn())dx >


(1 P2)(n())4dx


4n + 5 4n + 7
4n + 4 4n + 6 _1


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


1 x1 dx]
k=1 (X Xk)2


1 1- x 2
2 (1 i x2)2(P~(x))4 Z ( )2 dx
(2n + 2)2 _1 k=1 (X- k)

(4n + 5)(4n + 7) 1 [2'
8(n + 1)(2n + 3) 1J_,1 (X))4dx


(4n + 5) f -
4(n + 1)(2n + 3)2 j k=1


1 X2
(x- L)2


n 1 2
(1 x2)2(p())4 ( k2 d
k=1k)l


(4n + 5)(4n + 7)/(1 X ))
-8(n + 1)(2n + 3) -,1


1
(2n + 2)2








(4n + 5) 1
4(n + 1)(2n + 3)2 (2n + 2)2


1 2z.
1 k=1 k


2n2 1 -1 24 4n(n +1 ) 11 -- 4 k2dZ
2n 2 (1 x2)(P(X))4d 4( + 1) X-2)(p (xxk)2dx
2n+ (2n + k=l


2n2
- (2n + 1)


(4n + 5)(4n + 7) (1 x2)3p()4dx
8(n + 1)(2n + 3) _1


r[4n(n + 1) 2n2 (4n + 5) 21
(2n + 1)2 (2n + 1) 4(n + 1)(2n + 3)2 (2n + 2)2

(1 )(P,())4 ( k)2
1 k=1 k


and from (5.3.3) we deduce


1 ) (X))4 3n3(4n + 5)(4n + 7) (1 2)3(pn())4d.
1(-21 > 64(2n + 1)(n + 1)(2n + 3) -


This proves Theorem 5.2. Now, as

(1 x)(P'(x))4dx > (1 x2)2(P())4d > (1 x2)3(())4dx
1 --1 1
we deduce (5.1.1) from (5.3.3). Similarly, we deduce (5.1.2) from (5.3.4) and (5.3.5).
This proves the Corollary. D



Proof of Theorem 5.3
Let P, E Hn and p > 2 even. Denote

= J_'
11-


Thus,








From Identity 5.2.4 we see that


pJr > ((p r )n (p- r 2))Jr+2


(r = 0,2,...,p- 4)


which yields


f1,(P'(x))Pdx
f, (P,(x))Pdx


Jo
Jp


Jo J2 Jp-4 Jp-2
\-2 J4 Jp-2 Jp


> (p )n-(p-2)
\ ~p /


((p -3)n- (p -4)
p


3n-2
*\ p~


((p- 1)n (p 2))((p 3)n (p 4))...(3n 2) 2 f2 1((())) d
P-1 1 (P()) 2
P2 P f P ()) d2


pn(pn + 1)
4(p 2)
4(pn 2)


pf+1
P 2


((p- 1)n -(p- 2))((p 3)n (p 4))...(3n- 2)n(pn + 1)
pt(pn 2)


with the last inequality following from Theorem 1.4.4. Now, we use that


St-1'(1 t)-ldt= r(p)F(q)
0 r(p + q)'


where p and q are positive real numbers. To see that the result above is asymptotically
sharp, we note that for the polynomial Po(x) = (1 x2)m (n = 2m) we have


f ,(Po(x))Pdx
f (Po(x))Pdx


nP fl xP(1-2 dx
fJ,(1- 2)x7dx


f), (P(X))p-2 (P (Xn)2dx
f- (P.(x))Pdx









nP fot 2 (1 t) 2 dt
fo' t t)-Ta
S-t) F2 (P+3)dt
nP y E(-3) ( ... (2) (.1) F (222) ( )



2 22
np(np-2)...(np-(2p-2))
2P


p(np + 1)(np 1)-- (np (p 3))
- n"(p 1)(p 3)...5 -3
np(np 2) (np (2p 2))


This proves Theorem 5.3.













CHAPTER 6
SUMMARY AND CONCLUSIONS

6.1 Synopsis

The problems discussed in this dissertation come from the vast field of Approx-

imation Theory. The areas of this field considered here are Birkhoff Interpolation,

Erdos-type inequalities and Turan-type inequalities.
In Chapter Two, we proved the existence and uniqueness in the case of the

'modified' (0, 1,3,4) case of Birkhoff Interpolation on the zeros of 7r,i(x), and then

provided an explicit representation in this case. Chapter Three presents the esti-

mates of the fundamental polynomials, which are then used to prove the uniform

convergence of the 'modified' (0, 1,3,4) Birkhoff-Fej6r operator for the entire class of

continuous functions on the interval [-1,1]. This provides only the second known

case of such a Birkhoff-Fejer operator.
In Chapter Four, a classical theorem of P. Erdos [10] is extended in the L2

norm with the ultraspherical weight w(x) = (1 x2), a > -1, for the Lorentz class

L, of polynomials. This result also generalizes some previously known partial results

in the L2 norm. Then the result of Erd6s is extended in the L4 norm with weight

w(x) = (1 x2)3. These results are the best possible, as the inequalities given are

sharp. We note that this is the first sharp result extending the inequality of Erdos
in the L4 norm.

In Chapter Five, a classical theorem of P. Turn [45] is extended in the L2 norm
with ultraspherical weight w(x) = (1 X2)1, a > 1 (a > -1 if Pn(l) = 0). Next, we

extended the inequality of Turan in the L4 norm with weight w(x) = (1 x2)3. We

note that this is the first sharp extension of the inequality of Turan in the L4 norm.








These results provide lower bound analogues to the results given in Chapter Four

for the class Hn of polynomials. These results are again the best possible, providing

inequalities that are sharp. Finally, the classical theorems of P. Turan (in the uniform

norm) and A.K. Varma [51] (in the L2 norm) are extended in the LP norm for p an

even integer. This result is sharp in the asymptotic sense.

6.2 Open Problems

We close this chapter by noting some open problems related to those presented

in this work. First, we recall that the case of the 'modified' (0, 1,3,4) interpolation

has been generalized into the 'modified' (0,... ,r 3, r 1,r) case. Similarly, the

'modified' (0,2), and the 'modified' (0,3) and (0,1,4) cases have been extended to the

'modified' (0,..., r 2, r) and (0,..., r 3, r) cases, respectively. In the paper of

A. Sharma, J. Szabados, A.K. Varma, and the author [38], the problem of existence

and uniqueness is settled, and explicit forms are provided. We note that the explicit

representation of the last fundamental polynomials is found in the same manner as

that given here. However, the remaining fundamental polynomials are determined in

a different manner, one that is more amenable to the general situation. Nevertheless,

better estimates are obtained from those given in this work. One expects that the

(0, 3) and (0, 1,3, 4) cases are the only two of all the previously studied cases which
converge uniformly for the entire class C[-1, 1]. These two cases have a special

'balanced' nature.

One problem would be to show that the given error of 0 w (f, log)) is sharp,

and if not, to provide a sharper estimate. While it is easy to write down an explicit

representation in the 'pure' (0, 1,3,4) case, having handled the 'modified' case, this

representation is too complicated to be useful as a means of estimating the fundamen-

tal polynomials in the 'pure' case. Thus, convergence results cannot be determined.

Hence, another problem would be to find an even simpler explicit representation in

the 'pure' (or 'modified') (0, 1,3,4) case, and then provide convergence results in the








'pure' case. One would expect the error to be O(w4(f, I)). Another problem

would be to find other Birkhoff-Fej6r operators which also converge uniformly for all
continuous functions on the interval [-1, 1].
Concerning Chapter Four, it would be interesting to provide the sharp inequal-

ity in the L4 norm when w(x) = 1, and in the more general case w(x) = (1 X2),
a > -1, as well as providing sharp results in the LP norm for larger values of p.
Also interesting is the extension of these results to higher derivatives. For the related
extensions already known in the uniform norm, see W.A. Markov [25] (for all higher
derivatives of algebraic polynomials of degree n) and J.T. Scheick [36] (for first and
second derivatives of polynomials in the Lorentz class).
In Chapter Five, again it would be interesting to provide the sharp inequality

in the L4 norm when w(x) = 1, as well as when w(x) is the ultraspherical weight.
Also, it would be interesting to provide the sharp inequality in the LP norm, p even,
for each n, and also for higher derivatives. For the known results in the uniform norm
for the class H,, see J. Er6d [12] (for the sharp result in the case of the first derivative)
and V.F. Babenko and S.A. Pichugov [4] (for the case of the second derivative).













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BIOGRAPHICAL SKETCH


Brandon Underhill was born in Jacksonville, Florida, in August, 1969. He

received his Bachelor of Arts degree with a minor in Classics from the University

of Florida in August of 1990. He entered graduate school as a teaching assistant at

the University of Florida that same month. He received his Master of Science degree

from the University of Florida in August of 1992. He delivered an invited talk at

the 1995 International Conference on Approximation Theory and Function Series in

Budapest, Hungary, in August, 1995.







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.


/Joseph lover Chairman
ofessor 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.
S&/zK
Li C. Shen
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.


-ermit N. Sigmon V
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.


Stephen A. Saxon
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.


Pejaver V. Rao
Professor of Statistics

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.

May 1996
Dean, Graduate School





















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