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Error bound for polynomial and spline interpolation 

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Howell, Gary Wilbur, 1951 

Copyright Date: 
1986 
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Full Text 
ERROR BOUNDS FOR POLYNOMIAL AND SPLINE INTERPOLATION
By
GARY WILBUR HOWELL
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
1986
Copyright 1986
by
Gary Wilbur Howell
To my wife, Nadia
ACKNOWLEDGEMENTS
I wish to express my sincerest appreciation to Dr.
Arun Varma for his research counseling and assistance
throughout my graduate school years. I wish also to thank
Drs. David Drake, Nicolae Dinculeanu, and Soo Bong Chae,
for their teaching and for encouraging me to pursue the
doctorate in mathematics, as well as Drs. Vasile Popov and
A. I. Khuri for their kindness in serving on my committee.
Finally of course, my parents and wife deserve rather more
thanks than can be easily expressed.
L
TABLE OF CONTENTS
Page
* iv
* 711
ACKNOWLEDGEMENTS . . . . *
ABSTRACT . . . . . *
CHAPTER
INTRODUCTION
ONE
Lagrange and HermiteFejbr
Interpolation ....
Optimal Error Bounds for Two Point
Hermite Interpolation ...
Birkhoff Interpolation ....
Polynomial Approximation ....
Spline Approximation .....
Parabolic Spline Interpolation ..
Optimal Error Bounds for Cubic
Spline Interpolation ...
. 2
. 4
. 8
. 13
.. .16
. 25
. 27
TWO
BEST ERROR BOUNDS FOR DERIVATIVES OF TWO
POINT LIDSTONE POLYNOMIALS....
Introduction and Statement of
Main Theorem .......
Preliminaries.......
Proof of Theorem 3.1......
A QUARTIC SPLINE
THREE
Introduction and Statement of
Theorems ...
Proof of Theorem 3.1 ...
A QUARTIC SPLINE ....
Introduction and Statement of
Theorems ....
Proof of Lemma 4.1 ....
Proof of Theorem 4.1 ...
Proof of Theorem 4.2 ...
. 43
. 54
. .
. .
. 60
FOUR
. .
.
.
1
FIVE IMPROVED ERROR BOUNDS FOR THE
PARABOLIC SPLINE . . . 81
Introduction and Statement of
Theorems . . . 85
Proof of Theorem 5.1 . . . 85
Proof of Theorem 5.2 . . . 88
Proof of Theorem 5.3 . . . 98
SIX CONCLUDING REMARKS .. ... 107
REFERENCES . . . . . 110
BIOGRAPHICAL SKETCH .. .. .. 113
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
ERROR BOUNDS FOR POLYNOMIAL AND SPLINE INTERPOLATION
By
Gary Wilbur Howell
August 1986
Chairman: Dr. Arun K. Varma
Major Department: Department of Mathematics
The present dissertation is motivated by a desire to
have a more precise knowledge of asymptotic approximation
error than that given by best order of approximation. It
owes its inspiration to a paper by G. Birkhoff and A. Priver
concerning error bounds for derivatives of Hermite
interpolation and a paper of C. A. Hall and W. W. Meyer
concerning error bounds for cubic splines.
In Chapter One we consider well known results
concerning interpolation, polynomial approximation and
error analysis of spline approximation. The results given
here are meant to provide a context for the theorems given
in later chapters. In Chapters Two and Three we consider
the problem of best error bounds for derivatives in two
point Birkhoff interpolation problems.
Vll
Chapter Four presents the problems of existence,
uniqueness, explicit representation, and the problem of
convergence for fourth degree splines. Moreover we also
consider the problem of optimal pointwise error bounds for
functions f 8 C(5) [0,1]. In Chapter Five our main object
is to sharpen the error bounds obtained earlier by Marsden
concerning quadratic spline interpolation. By doing so we
obtain in some special cases error bounds that are in fact
optimal.
viii
CHAPTER ONE
INTRODUCTION
The purpose of this chapter is to provide a context
for the results derived in succeeding chapters. In order
to show some of the important achievements in
approximation by polynomials, we discuss briefly the
Lagrange and HermiteFejek interpolations, which match a
given function at any finite number of distinct points.
After exploring the question of computational stability of
a given interpolation, we discuss in some detail the
problem of best order of approximation by polynomials as
initiated by S. N. Bernstein [1912], D. Jackson [1930],
and A. Zygmund [19681.
In contrast to high order approximation by a single
polynomial, we next consider in great detail the problem
of approximating a given function f(x) defined on [a,b]
by the interpolatory piecewise polynomials known as
splines. Special attention is given to the problem of
approximating by piecewise cubic and piecewise parabolic
splines. The study of these splines motivates us to also
study two point Hermite and Birkhoff interpolations.
L
Lagrange and HermiteFej~r Interpolation
Let X denote an infinite triangular matrix with all
entries in [1, 11
x0,0
(1..1) X: x0,l x1,1
x0,n x1,n ..x .n
We denote by Ln[f,x;X] the Lagrange polynomial of
interpolation of degree 4 n which coincides with f(x) in
the nodes xkn (k = 0, 1, ., n). Then
(1..2) Ln [f,x;X] = f(xkn)1kn(x)
k=0
where
(1.1.3) 1k (x) = n(x)
[x xkn) a '(xkn)
wn(x) = H (x xkn *
k=0
It is known from the results of G. Faber and S. N.
Bernstein that no matrix X is effective for the whole
class C of functions continuous in [1, 1]. Bernstein
showed that for every X, there exists a function f0(x) and
a point x0 in C[1,11 such that
(1.1.4) limn> m L [f0,x0;X] = m
L. Fej~r [1916] showed that if instead of Lagrange
interpolation, we consider the HermiteFejer interpolation
polynomials, the situation changes. The HermiteFejhr
polynomials Hn+1[f,X,X] are of degree L2n + 1 and are
uniquely determined by
L
(1.1.5) H[f,xkn;X) f(xkn), Hn+1) '"kn;X1 = 6kn
where 6kn are arbitrary real numbers, k = 0, 1, . n.
The explicit form of Hn+1[f,x;X] is given by
n n
(1..6)H .[f X; X] = 1 f(xkn) hkn(x) + C kn kn (x)
k=0 k=0
where
(1..7)hkn(x) = {1 an" (xkn)(x xkn) } Ikn2(x)
Wn (xkn)
=: kn(x) Ikn2(x)
and
(1.1.8) kn(x) = (x xkn) 1kn2(x).
Fejk brought out the importance of Hermite interpo
lation by introducing the concept of "strongly normal"
point systems. To each set of n + 1 distinct points x0'
xl, . xn, Fejbr associates a set of n + 1 points X0'
X1' * 'Xn which are the zeros of the linear functions
k(x). The points XO' X1, . ., X, are said to be the
conjugate point system of x0, xl, ., xn. A system of
points x0, xl, ., xn is called strongly normal if the
conjugate point system lies inside [1, 1]. For example,
the zeros of the Tchebycheff polynomial Tn(x) = cosn9,
cose = x form a strongly normal point system. Fe j r
proved (using these ideas) that HermiteFejkr interpola
tion polynomials based on strongly normal point systems
(and under certain conditions on Skn) converge uniformly
to f(x) on [1, 11.
Optimal Error Bounds for Two Point Hermite Interpolation
In order to motivate the present day work on error
bounds, we first consider the classic error bound of
Cauchy. Let us consider once more the interpolation
formula of Lagrange. Let f(x) e C[a,b] and consider the
Lagrange interpolation polynomial
Ln [f,x] = f(xkn) 1kn(x).
k=0
Next we set
(1..1) e(x) = f(x) L, [f,x] .
In the case f(x) is itself a polynomial of degree
I n, then it is easy to see from the uniqueness of the
Lagrange interpolation polynomial that e(x) = 0. Thus it
is of interest to study what can be said about e(x) if
f(x) is a given smooth function other than a polynomial of
degree I n. The following theorem gives the most widely
known error bound.
Theorem 1.1 (Cauchy). Let f(x) e C[a,b] and suppose
that f(n)(x) exists at each point of [a,b]. Let L [f,x]
be the element of the class of polynomials of degree
I n 1 that satisfies the equation
(1.2.2) L [f,xin] = f(xin) i=0 ,..,n
Then for any x in [a,b], the error
e(x) = f(x) Lnlf,xJ
has the value
(1.2.3) e~x) = w,(x) f(n+1)(5 )/(n+1)!
where E is a point of [a,b] that depends on x and
I
W,(x)=C (xxin)*
i=0
An immediate consequence of (1.2.3) is the inequality
(1.2.4) le(x)l
where I Idenotes the supremum norm on [a,bl. If we set
f(x) = W (x), we see that (1.2.4) becomes an equality.
Thus the right hand side cannot be made smaller. We
therefore say that (1.2.4) is an optimal bound.
The Equations (1.2.3) and (1.2.4) have been
extensively studied. For instance, the study of
minimizing I~ wn led to Tchebychev's system of
orthogonal polynomials. For a good discussion of some of
the elementary analysis associated with this error bound,
see Powell [1981].
In contrast to the precise and beautiful pointwise
Cauchy bound, very little has been known about precise
polynomial derivative errors. Denoting e(x) as the Cauchy
remainder for Lagrange po lynomial interpolation, we
consider the role played by the term f(n+1)(S). If f e Pn
(the class of polynomials of degree < n), the remainder
vanishes identically. For a fixed x, we may consider the
remainder
e (x) = f(x) L [f,x]
as a process which annihilates all elements of Pn. We
may now formulate the following theorem of Peano [19131.
Theorem 1.2 (Peano). Let L be a continuous linear
functional such that L(p) = 0 for all p e Pn. Then for
all f f C(n+1)[a,b],
(1.2.4) L(f) = J f(n+1)(t) K(t) dt
where
K(t) = ( LX[(x t)n] 3 n
and
=(x t)n for x > t
(x t)f
= for x < t.
The notation Lx[(xt)n] means that the functional L
is applied to (x t)n considered as a function of x.
For a detailed study of the Peano theorem we refer to P.
J. Davis [1975] and to A. Sard [1963]. We next turn to an
application of the Peano theorem to derive pointwise
optimal derivative error bounds.
Let u(x) e C 4)[0, h] be given; let v3(x) be the
unique Hermite interpolation polynomial of degree < 3
satisfying
(1.2.5) v3(0)=u(0) v3 (h)
v'3(0)=u'(0) v'3 (h)=u'(h).
Ciarlet, Schultz and Varga [1967] obtained a
pointwise error bound for e(x)= v3(x) u(x) and its
derivatives in terms of
U =max0
Their bounds are
(1.26) ~ k)()] hk[x(hx)]2 U k = 0, 1, 2.
k! (4 2k)!
For k = 0, (1.2.6) is best possible, since equality
holds for u(x) = x2(hx)2, whose Hermite interpolation
polynomial is v=0.
G. Birkhoff and A. Priver [1967] obtained the
following optimal error bounds on the derivative le(k)(x)l
in terms of U.
Theorem 1.3 (Birkhoff and Priver). Let u(x) 6 C [0,1].
Then we have (h = 1)
(1.2.7) le'(x) /U L x(x1)(2x1) ] / 12
for 0 < x < 1/3 ,
S[ 16x3 105x2 + 197x 162
+ 66/x 13/x2 + 1/x3 ] / 96
for 1/3 < x < 1/2.
(1.2.8) le"(x)l/U L [ 48x5 + 24 103
+ 54x2 12x + 1 ] / 2(1x)3
for 0 < x < 1/3 ,
< ( 6(x1/2)2 + 1/2 ] / 12
for 1/3 < x < 2/3.
(1.2.9) le"'(x) /U ( (x1/2)4 +3x12)/+ 3/16
for 0 < x< 1.
For 1/2 < x < 1 the bounds of e(k)(x) are given by
(1.210)e(k) (x) = e(k) (1x) k = 0 1, 2, 3.
Further, from Birkhoff & Priver, the uniform error
bounds are given by
le(r)(x) < ar U r = 1, 2, 3,
g r
424!
(1.2.11) al = (/Ti)/216
a2 = 1/12
a3 = 1/2.
The proof of the above theorem is based on the Peano
kernel theorem. It gives a general and highly useful
method for expressing the errors of approximations in
terms of derivatives of the underlying functions of the
approximation. For a computer routine which gives
polynomial error bounds by numerical quadrature of the
Peano kernel, see Howell and Diaa [1986]. Stroud [1974]
gives a readable account of some other applications.
Birkhoff Interpolation
We have just observed that in problems of Hermite
interpolation, function values and consecutive derivatives
are prescribed for given points. In 1906, G. D. Birkhoff
considered those interpolation problems in which the
consecutive derivative requirement can be dropped. This
more general kind of interpolation is now referred to as
the Birkhoff (or the lacunary) interpolation problem (s) .
The Birkhoff interpolation problem differs from the
more familiar Lagrange and Hermite interpolation in both
its problems and its methods. For example, Lagrange and
Hermite interpolation problems are always uniquely
solvable for every choice of nodes, but a given Birkhoff
interpolation may not give a unique solution.
More formally, given n + 1 integer pairs (i,k)
corresponding to n + 1 real numbers cilk, and m distinct
real numbers xi, i. = 1, 2, , m n + 1, a given problem
of polynomial interpolation is to satisy the n + 1
equations
(1.3.1) P (k)(x ) F~
with a polynomial P, of degree at most n. (We are using
the convention that Pn(0)(x) = Pn(x).)
If for each i, the orders k of the derivatives in
(1..1)form an unbroken sequence k = 0, 1, . ,ki, then
the interpolation polynomial always exists, is unique, and
can be given by an explicit formula. If some of the
sequences are broken, we have Birkhoff interpolation. As
remarked by Professor Lorentz [1983], the two cases are as
different as, let us say, the theory of linear a~nd
nonlinear differential equations.
Pairs (i,k) which appear in (1.3.1) are most easily
described by means of the interpolation or incidence
matrix E. If P (k)(xi) is specified in (1.3.1), we put a
"1" in the i+1st column and kth row of E. If P,(k) (x ) is
not specified in (1.3.1), then a "O" appears in the i+1st
column and kth row. Each of the m rows of E has a non
zero entry. An incidence matrix E and a pointset X, which
lists the points xi, specify a Birkhoff interpolation
problem of the type of (1.3.1). For a given E and X, the
unique existence of an interpolation polynomial of degree
n + 1 is equivalent to the invertibility of the system of
equations given by (1.3.1), or equivalently to the inver
tibility of a matrix V which we will refer to as a
generalized Vandermonde matrix V. For Lagrange
interpolation of the points xi, i = 1, 2, ., n + 1,
the Vandermonde V is given as
1 1. .. 1
x1 x2 . xn+1
(1.3.2) V =. ..
n n n
xl x2 ..xn+1
Inversion of the Vandermonde gives the coefficients of the
fundamental functions 1kn(x) of Lagrange interpolation.
As Lagrange interpolations are always unique, it follows
that Vandermonde matrices are invertible.
For a given system (131,it is not hard to
construct an analagous matrix to (1.3.2), which we will
refer to as the generalized Vandermonde. Just as
inverting the Vandermonde matrix gives the fundamental
functions of Lagrange interpolation, inverting the genera
lized Vandermonde gives a convenient form for representing
a Birkhoff interpolation. The Vandermonde and its
counterpart for Birkhoff interpolation are examples of
Gram matrices, of which a good account is to be found in
Davis [1975].
Though invertible, the Vandermonde matrices are known
to be extremely illconditioned for realvalued
interpolation. Many of the generalized Vandermonde
matrices associated with Birkhoff interpolation processes
are much better conditioned, illustrating an advantage of
Birkhoff interpolation over the more traditional Lagrange
interpolation. To make this point more explicit, we
define "condition" of a matrix.
For a given norm I and invertible matrix M, we
define the condition cond (M) of the matrix M by
(1.3.3) cond(M) = II10 IM11
If we rescale the Birkhoff interpolation problem
specified by E and X to the unit interval, we can define
the conditionof an interpolation as the conditionof the
associated generalized Vandermonde. In the L2 norm for
eleven equally spaced points, the condition number of
Lagrangian interpolation is on the order of a million. On
the other hand, Lagrangian interpolation on eleven equally
spaced complex roots of unity has L2 condition number one,
as does the eleven term MacLaurin expansion.
Computationally speaking, the inverse of the
condition number of a matrix M is the norm distance of M
from a singular matrix (See Golub and Van Loan [1983]).
For example, the Vandermonde for Lagrange interpolation of
eleven points on the unit interval is thus seen to be a
norm distance of only onemillionth from being singular.
Not only is the illconditionedness of the Vandermonde
troublesome in determining the coefficients of the
fundamental functions, but it also causes problems of
roundoff error in evaluating a polynomial by use of the
fundamental functions. For these reasons, it is very much
preferable to use a wellconditioned interpolation.
The MacLaurin expansion, having diagonal generalized
Vandermonde, is as wellconditioned as is possible.
Another particularly wellconditioned interpolation is the
Lidstone interpolation.
A Lidstone polynomial is a truncation of a Lidstone
series. In turn, a Lidstone series is a generalization of
a Taylor series which approximates a given function in the
neighborhood of two points instead of one. Such series
have been studied by G. J. Lidstone [1930], by Widder
[1942], by Whittaker [1934] and by others. More
precisely, the series has the form
(1.3.3) f(x) = f(1)AO(x) + f(0)AO(1x) + f" (1)A1(x) +
f" (0)Al(1x) +...
where A (x) is a polynomial of degree 2n + 1 defined by
the relations
Apn(x) = x
(1.3.4) n," (x) = An1l(x)
An(0) = A (1) = 0, n = 1, 2,...
Thus it is clear that the sum of an even number of
terms of the series (1.3.3) is a polynomial which coin
cides with f(x) at x = 0 and at x = 1. Moreover, each
even derivative of the polynomial coincides with the
corresponding derivative of f(x) at those points.
Polynomial Approximation
Weierstrass first enunciated the theorem that an
arbitrary continuous function can be approximately
represented by a polynomial with any degree of accuracy.
We may express this theorem in the following form.
If f(x) is a given function, continuous for
a < x < b, and if E is a given positive quantity, it
is always possible to define a polynomial P(x) such that
(1.4.1)f(x) P(x)] < E
for all a < x < b.
It is readily seen that the number of terms required
to yield a specified degree of approximation, or under the
converse aspect, the degree of approximation attainable
with a specified number of terms, is related to the
properties of continuity of f(x). Naturally this has led
to many interesting developments in the theory of degree
of approximation of continuous functions by polynomials to
which we turn to describe.
A first important step in building this theory was
made by D. Jackson [1930]. Let f 8 C[1,1]. Suppose that
we define the best approximation of f by polynomials of
degree n by
(1.4.2) E (f) = inf If P,
where Pn ranges over all algebraic polynomials of degree n
and Ifl  = max If(x)l, a
problem of estimating E (f). To describe his results we
need the following definition.
Definition 1.1 If f e Cla,b], then the modulus of
continuity of f is a function (f,h) such that
(1.4.3) (f,h) = sup xylJh; x,y 6 [a,b] /f(x) fly) .
Now Jackson's theorems may be easily stated.
Theorem 1.4 (Jackson). Let f be continuous on [1,11.
There is a positive constant A such that
(1.4.4) En(f) I A w(f,1/n) n = 1, 2,..
where A is independent of f.
An important corollary of Theorem 1.3 deserves to be
mentioned. Let Lipa [1,11(M) (or simply Lipa) be the
class of functions f in C[1,1] such that
(f(x) f(y)l I M IxyJa
for all x and y in [1,1]. It is easy to see that
f 6 Lipa[1,11(M) if and only if
w(f,h) 0
We then have the following consequence of Jackson's
theorem.
Corollary 1.5 Let 0 < a< 1. If f e Lipcl_ l(M), for
some constant M, then
(1.4.5) En(f) < A for n = 1, 2,...
for some positive constant A.
A. F. Timan [1951] noticed the following
strengthening of Jackson's theorem.
Theorem 1.6 (Timan). There is a positive constant C such
that if f e Cl1,11 and n is a natural number, then there
is a polynomial Pn of degree n such that
(1.4.6) If(x) P (x)l ( A[ w(f,J1 x2) + w(f,1/n2)
for all x in the interval [1,11.
In this result, in contrast to the theorem of
Jackson, the position of the point x in the interval
[,]is taken into consideration and it is apparent that
for the polynomial P (x) thus constructed, as Ixl > 1,
the deviation If(x) P (x) is of magnitude w (f,1/n2)
Following the important theorem of Timan, V. K.
Dzjadyk [1956] proved the converse of Jackson's theorem.
Theorem 1.7/ (V. K. Dzjadyk). Let f 8 C[1,1]. Suppose
that: 0
each n there corresponds a polynomial P, of degree n such
that
(1.4.7) If(x) Px)
n n2
if and only if w (f,h) ( C he1 for some constant C.
From Jackson's theorem we noticed that if f 8 Lipa,
then
E (f) < AM n = 1, 2...
where A is an absolute constant. To achieve a more rapid
decrease to 0 of En(f), it is necessary to assume more
smoothness for f, for example, that f has several
continuous derivatives. Let Cr[1,11 r = 0, 1 .
denote the subset of C[1,1] consisting of those functions
which possess r continuous derivatives on [1,1]. For
this class of functions, Dunham Jackson proved also the
following direct theorem.
Theorem 1.8 (D. Jackson). If f e C(r) [1,1], then
(1.4.8) En(f) I Ar (1/n)r ,(f(r),1/n) n = 1, 2,...
For many important contributions we refer to the work
of G. G. Lorentz [1983].
Spline Approximation
One uses polynomials for approximation because they
can be evaluated, differentiated and integrated easily and
in finitely many steps using just the basic arithmetic
operations of addition, subtraction and multiplication.
But there are limitations of polynomial approximations.
For example, the polynomial interpolant is very sensitive
to the choice of interpolation points. If the function to
be approximated is badly behaved anywhere in the interval
of approximation, then the approximation is poor every
where.
This global dependence on local properties can be
avoided when using piecewise polynomial approximation.
Concerning piecewise polynomial approximation, Professor
I. J. Schoenberg remarked that "polynomials are wonderful
even after they are cut into pieces, but the cutting must
be done with care. One way of doing the cutting leads to
the socalled spline functions" (Schoenberg [1946],
p. 46).
Splines were introduced by Prof. Schoenberg in 1946
as a tool for the approximation of functions. They tend
to be smoother than polynomials and to provide better
approximation of low order derivatives. Though we will
later use the word spline in a somewhat broader context,
we first give the more traditional definition.
Let
(1..1) xl < x2 < . < xk
be a sequence of strictly increasing real numbers called
the knots of the spline function. We may say sm(x) is a
spline function of degree m having the knots
xl, x2,. xk
if it satisfies
a) sm(x) e Cm (1 poo)
b) In each interval (xi, xi+1), including ( ,xl)
and (xk'm), the restriction of sm(x) to (xi, xi+1) is a
polynomial of degree at most m. Thus, a step function
s0(x) may be regarded as a spline function of degree 0,
while a spline function of degree 1 is a polygon (broken
line function) with possible corners at some or all of
the possible corners at some or all of the points (1.5.1).
Similarly, s2(x) has a graph composed of a sequence of
parabolas which join at the knots continuously together
with their slopes. Both for a smoother approximation and
for a more efficient approximation, one has to go to
piecewise polynomial approximation with higher order
pieces. The most popular choice continues to be a
piecewise cubic approximating function. Various kinds of
cubic splines are in use in numerical analysis. The ones
most commonly used are complete cubic splines, periodic
cubic splines and natural cubic splines.
A spline function of degree m with k knots is repre
sented by a different polynomial in each of the k+1
intervals into which the k knots divide the real line. As
each polynomial involves m + 1 parameters, the spline
function involves a total of (m+1) (k+1) parameters.
However, the continuity conditions stated earlier impose
certain constraints on those parameters. At each knot,
the two adjoining polynomial arcs must have equal
ordinates and equal derivatives of order 1, 2, . .,
m 1. Thus, m constraints are imposed. It is easy to
see that every spline function s(x) of degree m with the
knots xl, x2, ., xk has a unique representation in the
form
(1..1)s(x) = Pm(x) + q c (x x )m
]=1
where Pm(x) denotes a polynomial of degree m and
(1.5.2) x = xm x > 0
Also
(1.5.3) Cj = (1/(m)!) ( S(m)(x +) S ")(x 3.
The class of "natural" spline functions was intro
duced by Prof. Schoenberg [19461. A spline function s(x)
of odd degree 2p1 with knots xl, x2, ., xk is called
a natural spline function if the two polynomials by which
it is represented in the two end intervals ( ,xl) and
(xk,+ ) are of degree p1 or less. It is easy to express
the natural spline functions by
(1.5.4) stx) = Pp1(x) + Cj (xx ) 2p1
j=1.
where
C C x r = 0, r =I p,+1, ., 2p1.
j=1.
The following theorem states an important interpola
tion property of natural spline functions.
Theorem 1.9 Let (x yi ), i= 1, 2, .., k, be given
data points, where the xi's form a strictly increasing
sequence, and let p be a positive integer not exceeding n.
Then there is a unique natural spline function s(x) of
degree 2p 1 with the knots xi such that
(1.5.5) s(xi) = i ,2 ...,k
Natural spline functions possess certain impressive
optimal properties and can be shown to be the "best"
approximating functions in a certain sense. This is the
content of the next theorem.
Theorem 1.10 Let P(x) be the unique natural spline
function that interpolates the data points (xji'T '
i =1, .. ,kin accordance with Theorem 1.7. Let
f(x) be any function of the class C(P that satisfies the
conditions
(1.5.6) f(xi) = i ,2,..,k
Let (a,b) be a finite interval containing all the knots
xi. Then
(1.5.7) j [f 2) (x)]2 dx ji~gs ()(x) dx
a a
with equality only if f(x) = s(x).
The effectiveness of the spline approximation can be
explained to a considerable extent by its striking conver
gence properties. Interesting contributions were made by
J. N. Ahlberg and E. N. Nilson [1964], C. DeBoor and G.
Birkhoff [1964], A. Sharma and A. Meir [1967], M. J.
Marsden (1972], T. R. Lucas 11974], E. W. Cheney and F.
Schurer (1968], C. A. Hall [1968], C. A. Hall and W. W.
Meyer [1976], and A. K. E. Atkinson [1968]. As a good
reference on splines which offers a good comparison of the
approximating properties of polynomials and splines, we
recommend A Practical Guide to Splines by C. DeBoor
[1978].
First we discuss error analysis for the class of
functions f(x) 6 C(2) with period one. Let
(1.5.8) (xi =,0: 0 = xn0 < xn1 < ...
be a division of [0,11 of mesh gauge
(1.5.9) hn = max0
where
hni = xni xni1 *
A periodic cubic spline function yn(x) is .a function
composed of a cubic polynomial in each of the intervals of
{xi =,0 with the requirement that
y (x) e C(2)[0,11
and
yn () = yn () i = 0, 1, 2.
It was observed by Walsh, Ahlberg and Nilson [1962] that
there exists a unique periodic spline function yn(x) which
interpolates f(x) at the points xn,1. It was shown that
yn(x) and y'n(x) converge uniformly to f(x) and f'(x)
respectively as hn > 0. Later Ahlberg and Nilson [1966]
studied the more delicate question of the convergence of
y" (x) to f"(x). Writing
(1.510)An,i = hnli+1/(hnli + hn,i+1) '
i = 1, 2,..,k 1
and
n, = max0
where for m = kn' n,m+,1 is taken as
they show that
y" n(x) >f" (x)
uniformly provided that
h, > 0 and A, > 0
After this result, I. J. Schoenberg (1964a] raised the
question that it would be very interesting to find out to
what extent the condition A, > 0 is really necessary in
the above mentioned theorem. The above theorem together
with the open problem of Schoenberg lead to important
contributions by Birkhoff and DeBoor [1964], and Meir and
Sharma [1969] which we turn to describe.
In 1964, Garrett Birkhoff and Carl DeBoor made the
following contribution. Let f(x) e C'[0,1] and let
(1.5.11) Exi =0, O O = x0 < xl < .
be a partition. The function f(x) is now interpolated by
a cubic spline function s(x) (called a complete cubic
interpolation spline function) which means that s(x) is a
cubic polynomial when restricted to each interval
(xi'xi+1), and s(x) F C(2)[0,1]. Moreover s(x) is
uniquely defined by the conditions
f'(0) = s'(0) ,
f'(1) = s'(1).
This first important result concerning the error analysis
yielded the following theorem.
Theorem 1.11 Let f(x) 6 C(4)[0,11. Denote
e(r) = (r) s(r)
There are constants cr(m), r = 0, 1, 2, 3, depending
only on m > 0, such that
(1..1) e(r)(x) < c (m) h4r Il(r)
r = 0, 1, 2, 3,
provided that
mh < m,
hi = xi+1 xi
h = max hi
mh = [max(hi)]/[min(hi)l
and I I denotes the supremum norm.
The authors go a step further and prove a convergence
theorem related to f 6 C (3) [0,1].
Theorem 1.12 Le t f"' (x) be absolutely continuous on
[0,1]. Let (x @i=0,n (where k depends on n)beasqnc
of partitions of 10,1] such that hn = maxihi,n > 0 as
n > .Let mh,n I m as n > .Let e (x) be the error
incurred when f(x) is interpolated by a spline function on
{xi =,0,n. Then
le" 'n > 0
uniformly on [0,1] as n > "
The next important development came with some
interesting results by Prof. A. Sharma and A. Meir [1967]
concerning degree of approximation of spline interpola
tion. This paper does away with some annoying assumptions
under which uniform convergence of the interpolating cubic
spline and its derivatives was proven earlier (see above
for these restrictions).
Theorem 1.13 Let f(x) be continuous and periodic with
period unity. Let
(1.515)q = maxirj (hn,i/hndj)
where
hn,i = xn,i+1 xn,i
Let sn(x) be the cubic spline of period unity with
joints (or knots) xn,i, i = 0, 1, ., n in [0,1], such
that s (x) interpolates f(x) at the joints. Let
g1 = maxx g(x)] for g 6 C[0,1]
and
o(g,h) = max ( jglu) g(v) :uvl < h ), h > 0
The authors prove
i)
ii)
iii)
iv)
qn2) w(f,hn)
I f snj i (1 +
if f 8 C(1), then
If(r) srn
L 76 hn1rfn
r = 0 1
C(2), then
s (r)n 5 hn2r
C(3), then
s(r)nl C hn3r
if f e
lif(r)
if f 8
 f(r)
a (f",hnh)
r = 0, 1, 2 ;
n (f"',rhn) '
r = 0, 1, 2, 3 ;
where
C = 1 + q (1+qn 2
C = 1 + (1 + P )2/(2 P )
with
Pn = maxi (h~/ )for j=i1 +
satisfying
P, < 2.
From these results one can draw the obvious conclu
sions regarding uniform convergence of the interpolating
splines and derivatives. The arguments are surprisingly
simple. The uniform convergence of sn" to f", which
follows from iii), had been proved earlier by Ahlberg and
Nilson (see above) under the additional assumptions that
the mesh become eventually uniform, i.e.,
(1.5.20) lim,_>oo [hn,i/(hn,i+hn,i+1)] = 1/2.
Parabolic Spline Interpolation
Many interesting results were obtained by M. Marsden
[1974] concerning the approximation of functions by even
degree splines. Of particular interest are the simple
parabolic splines. If break points are the same as the
interpolated points, then the resulting spline is ill
behaved, as can be seen by simple examples (DeBoor
[19781). On the other hand, if we take the interpolated
points midway between break points, the parabolic splines
are very wellbehaved. In fact in the first theorem given
below, a good approximation to a continuous function is
assured with no conditions on the partition other than the
length of the largest subinterval being small.
We first give some necessary notation. Let
(1 .6 1) { i =0 0 = x 0 < x l < . < x n = 1
be a fixed partition of [0,1). Set
(1.6.2) h =xi xi1 h =max h,
zi = (xi + xi1)/2 ,
hO = hn ai = hi+1/(hi + hi+1) '
ci + ai = 1 for i = 1, 2, ., n.
Let
y e C[0,1] y(0) = y(1),
ly = sup { y(x) : 0 < x < 1 3
such that y is extended periodically with period 1.
A function s(x) is defined to be a periodic quadratic
spline interpolant associated with y and {xi 2=0
(1.6.3) a) s(x) is a quadratic expression on each
b) s(x) e C'[0,11
c) s(0) = s(1) s'(0) = s'(1),
d) s(z ) = y(zi) ,2,..,n
The following theorems were obtained by Marsden.
Theorem 1.14 (Marsden). Let {x} b prttono
[0,1], y(x) be a continuous 1 periodic function and s(x)
be the periodic quadratic spline interpolant associated
with y and (xi3 n=0'
Then
(1.6.4) Isi < 2 yl Ilsl (< 2 yff ,
leil 2 w(y,h/2) ,
lell L 3 w(y,h/2).
(where si = s(xi) and ei = y(xi) s(xi) *
The constant 2 which appears in the first of the above
equations can not, in general, be decreased.
Theorem. 1.15 (Marsden). Let y and y' be continuous 1
periodic functions. Then
(1.6.5) (js'i ( < 2 ly ,
le (h w(y', h/2) ,
leil
le  (5/4) h ly' ,
le'il I 3 o(y',h/2),
Je' J (9/2) w(y',h/2)
le II (13/8)h w(y',h/2).
Theorem 1.16 (Marsden). Let y, y', and y" be continuous
1 periodic functions. Then
(1.6.6) /e.l ( (1/8) h2 w(y",h),
le'i  (1/2) h w(y" ,h)
e'lr 2 h Ily" ,
lel  (5/8) h2 Ily
le" (x)j L [1 + (h/h )] w(y" h) ,
xi < x < xi+1
Theorem 1.17 (Marsden). Let y, y',r y", and y"' be
continuous 1 periodic functions. Then
(1.6.7) lej( ( 1/6 3 Jly"'
le'jl I (11/24) h2 Ily.
le"l L [hi + (2 h2/3 hil )]y" 
xi < x
Optimal Error Bounds for Cubic Spline Interpolation
An interesting application of the theorem of Birkhoff
and Priver (1967] (discussed above) was given by Hall
[1968] and subsequently by Hall and Meyer [19761, concern
ing optimal error bounds for cubic spline interpolation.
In order to describe these results let f e C(4) [0,11 and
let s(x) be the complete cubic spline function satisfying
the conditions (1.5.13). The main result of Hall and
Meyer may now be stated.
Theorem 1.18 (Hall and Meyer). Let s(x) be the unique
complete cubic spline interpolation satisfying(153)
Suppose
f 6 C(4)[0,1].
Then for 0 < x < 1
(1..1)f(r)(x) s(r)(x)l < c, h4r 11(4)
r = 0, 1, 2
with
h = max(xi+1 xi) c0 = 5/384 ,
cl = 1/24 c2 = 3/8.
Further, the constants c0 and cl are optimal in the sense
that
(1.7.2) c ( )r
r sup lf )rl
h4r l(4)
where the supremum is taken over all {xi ~=0 partitioning
[0,1] and over all f 8 C(4)[0,11 such that f(4) is not
identically equal to zero.
Varma and Katsifarakis (in press) were able to
resolve the cases of f e C(3) and f C C(2) in the
following theorems. Let s(x) be the unique complete cubic
spline satisfying the relationship:
(1.7.3) s(xi) = f(xi) i = 0, 1, ., k ;
s'(xi) = f'(xi) i = 0, k.
Theorem 1.19 If f, f', f", and f"' are continuous on
[0,1], then
(1.7.4) s()() r x
< cr h3r max0
r = 0, 1, 2
where
C0 = 1/96 + 1/27 cl = 4/27 ,
c2 = 1/2 + 4/ (3 J7).
Theorem 1.20 If f,f', and f" are continuous on [0,11,
then
(1.7.5) Is (r)(xf((x) x
where
a0 = 13/48 al = 5/6 a2 = 4.
CHAPTER TWO
BEST ERROR BOUNDS FOR DERIVATIVES OF
TWO POINT LIDSTONE POLYNOMIALS
Introduction and Statement of Main Theorem
Let 6 2ml0,h] be given and let v2m1 be the unique
Hermite interpolation of degree 2m 1 matching u and its
first m1 derivatives unj at 0 and h. Let e = v2m1 u
be the error function. For the special cases m = 2 and
m = 3, G. Birkhoff and A. Priver [1967] obtained
pointwise optimal error bounds on the derivatives e~k),
0 < k < 2m 1 in terms of h and max0
These results are described in detail in Chapter One.
Birkhoff and Priver note that for the cases m > 3, their
method is not likely to give analytically exact bounds,
though it can be adapted to give numerical approximations
to pointwise exact error bounds. In the next chapter, we
will directly apply the results of Birkhoff and Priver to
the case of u in C(2m)[0, h] and the interpolatory
polynomial w2m1 which matches u at 0 and h and which also
matches the 2nd through mth derivatives of u at 0 and h.
Analogously to using Hermite interpolation
polynomials, one may choose to approximate a given
function u(x) in C2ml0, h] by the socalled Lidstone
interpolation polynomial L2m1[u'x] of degree < 2m 1
matching u and its first m 1 derivatives u(2j) at 0 and
h. Thus L~m1[u'x] satisfies the following conditions
(where we assume h = 1):
(2..1)L~m1(2p) [u, 0] =u(2p)(0) ,
Lm1(2p) [u, 1] = u(2p) (1) ,
p = 0, 1, ., m 1.
The explicit formula for L2m1[u,x] is
m1
(2.1.2) LZm1[u'x] = u(2i)(1) i(x)
i=0
m1
+ C u(2i)(0) i(1x)
i=0
where
(21.) () 2i B (+) ,for i > 1
(213 i(X 2 2i+1(1x
(2i+1)! 2
and
(2.1.4) AO(x) = x.
Here B (x) denotes the Bernoulli polynomial
(2.1.5) Bn(x) = nk xk Bn
k=0
and where the constant Bj is given by
(2.1.6) Bj = ( ~) Bk 0 = 1.
k=0
That (2.1.2) in fact satisfies (1.)follows from
the facts
A 2p)(0) = 0 p =0,1 .. i ;
(2.1.7) A.;(2p)(1) = 0 p =0,1 ..,i 1
A 2i) (1) = 1.
The main object of this chapter is to obtain
pointwise optimal error bounds for
e j(x) =f j(x) L~m1 [ff~ x]
in terms of U = max0~
denotes the jth derivative of the Lidstone polynomial
defined by (2.2.2). An important role in Theorem 2.1 (see
below) is played by the Euler polynomial Q2m(x) of degree
2m given by the formula
(2.1.8) Q2m(x) = 0 G1(x't) Q2m2(t)dt m = 1, 2,..
where
(2.1.9) Q0(x) = 1
and
(2.110)G (x,t) = t (x 1) O < t < x < 1
=x (t 1), O< x < t <1
We may now state the main theorem as follows.
Theorem 2.1. Let u(x) e C m[0,11 and let L2m11u~x] =
L2m1(x) be the unique polynomial of degree L 2m 1
satisfying the conditions (2.1.1). Then, for 0 < x < 1,
with
U = max0
(2.1.11) lu(2j) (x) L2m12j)() (X U Q2m2j (x),
j = 0,1, . ,m 1
U Q2m2j(1/2)
and for j = 1, 2, ., m
'U ((12x) Q2m+22j'(x)
+ 2Qm+22j(x))
SU IQ2m+22j'(0)
where for a given integer k, Q2k(x) is the well known
Euler polynomial defined by (;2.1.7). Moreover, (2.1.11)
and (211)are both best possible in the sense that
there exists a function u(x) e C2m[0,11 such that (2.1.11)
and (211)become equality for every x e [0,11.
From (221)and (211)follow immediately the
also exact bounds
(2.113) u(2j) L~m 2j)I 2Qm2j (1/2)  u(2m)
and
(2.1.14) u(2j1) L2m12j1)
I Q2m+22j'(0)  Iu(2m)
where I I denotes the supremum norm on [0, 11.
Preliminaries
It is well known that the Bernoulli po lynomials
defined by (2.1.5) satisfy
(2.2.1) B '(x) = nBn1(x)
and
(2.2.2) Bn(1x) = (1)n Bn(x).
In particular it follows that
(2.2.3) B2n+1(1/2) = 0
From (2.2.1), (2.2.3) and (2.1.3)(2.1.6), we obtain
(2.2.4) A (x) = Ai1x i(0) = 0 A (1) = 0
The proof of Theorem 2.1 depends on repeated use of
the kernel G1(x,t) defined by (2.1.10). Let us consider
(2.2.5) g(x) = I G1(x,t) r(t)dt
x 1
= (x1)t r(t)dt + I (t1)x r(t)dt.
O x
On differentiating, we have
x
g'(x) = I t r(t)dt + (x1)x r(x)
(t1) r(t)dt x(x1) r(x)
x 1
= ft r(t)dt + I (t1) r(t)dt.
Differentiating once more with respect to x we obtain
(2.2.6) g" (x) = x r(x) (x1) r(x) = r(x).
Also
(2.2.7) g(0) = g(1) = 0
Let r(t) = m1()in (2.2.5). From the above
discussion it follows that
g(x) = i G1(x't) Am1(t)dt
satisfies
(2.2.8) g" (x) = Am1x) (0 g(1) = 0
From (2.2.4) we also know that for i 1
Ai (x) = Ai1 (x) A i(0) = 0 Ai(1) = 0
Therefore
(2.2.9) g~x) = Am(x) = I G1(xit) Am1(t)dt.
From (2.1.9) it follows that
(2.110)G (x,t) I 0.
Also A 0(t) = t > 0 O < t < 1 Therefore we
obtain from (2.2.9) that
(2..11 Alx) < 0, O
On using (2.2.9), (2.2.10), and (.1),we can assert
that
(2.2.12) A 2(x) > 0, O
Inductively, it follows that A (x) > 0 for 0 < x < 1
provided m is an even positive integer and AZm(x) I 0
O < x < 1 ifm is an odd positive integer. This property
of Am(x) will be needed many times in the proof of the
theorem.
The following iteratively defined kernels comprise
the essential machinery of the proof. Define
(2.2.13) G2(x~t) = JO G1(x~y) G (y~t)dt
and inductively
(2.2.14) G (x.t) = I Gl(x,y) Gn1ytd n = 2, 3,..
O,(itd
From (2.2.10) and (2.2.13) it follows that
(2.2.15) G (x,t) > 0 G (x,t) < 0;
0 < x < 1, O < t < 1.
In general
(2.2.16) (1)nGn(x,t) 0 ;
0 < x( < O < t < 1.
Finally, let us define
(2.2.17) h(x) = Gn(x,t) qlt)dt.
We note again that h(x) uniquely satisfies
h(2n) (x) = q(x)
(2.2.18)
h(2k)(0) = h(2k)(1) = 0 k = 0, 1, ., n1
We also need some of the known properties of Euler
po lynomials introduced in (2.1.7) and (218.We can
easily verify that
(2.119) 2n'(x) = Q2n2(x)
Q2n(0) = Q2n(1) = 0
Furthermore,
Q(2p) (0) = Q2n2p)(1) = p = 0, 1, ., n1,
(2.2.20) Q2n) n1) 2n") (0) =()
Q2n2j)(x) = (1)j Q2n2j (x)
Using (2.2.13) we note that
1
Q2x G1(x't) dt lty yid
O
Q4(x = j G1(x,t)d Q2t)d
and in general,
1
JO Gm (x,t) dt.
(2.2.21) QZm(x) = (1)"
Explicitly some of the first Euler polynomials are
given by
Q2(x) =x(1x) ,Q4(x) = x2(1x)2+x(1x)
2! 4!
06(x) = x3(1)+xx) x)+3x(1x).
61
Proof of Theorem 2.1
Le 2m1 denote the class of polynomials of degree
S2m1. Following the notation used by Birkhoff and
Priver [1967] we shall denote
81 DGm(x,t)
=u(x) for u(x) e P2m1 it follows
that for u e C2ml0,11
 2m1[u'xI
(2.3.1) Gm'(x
Since L~m1[~]
from the Peano theorem
(2.3.2) e(x) =: u(x)
= JGm(x,t) u(2m)(t) dt
where Gm(x,t) is the Peano kernel defined by (2.1.10) and
(2.2.14) Differentiating (2.3.2) we have
(2.3.3) e(2j)(x) = u(2j)(x) L~m12j)[u,x]
= JG (2j,0)(x,t) u(2m)(t) dt.
Let us substitute u(x) = Q2m(x) (as defined by
(2.1.7)) in (2.3.3) and use various properties as given by
(2.2..20) and (2.2.21). We then obtain
(2.3.4) Q2m2j) (x) L~m12j) Q2m'x]
= J G~j,0)x~t)2m!2m) (t)dt.
We know from (2.2.20)
(2.3.5) Q2m m(t) =Q2m~p()=(1
Moreover,
(2.3.6) Q2m2P)(0) = Q,2mp)(1) = 0, p = 0, 1, ., m1
It follows that
(2.3.7) L2m1 Q2m'x] = 0
identically. Thus (2.3.4) can be rewritten as
1
I Gm2j,0) (x,t) dt.
O
(2.3.8) Q2m2j)(x) = (1)m
Next we note from (2.2.14) that
Gm2,0) = Gm1(x't)
Hence
Gm4,0)(x,t) = Gm2,0)(x,t) =Gm2xt
and in general,
(2.3.9) G 2j,0)(x~t) = G (x,t).
From (2.2.16) and (2.3.9) we have
(2.3.10) (1)""3 Gm2j,0)(x,t) = (1)""j Gmj(x~t) > 0
in the unit square 0 < x < 1 O < t < 1.
Combining (2.3.3) (2.3.9) (2.3.10) (2.2.19) and
(2.3.8), it follows that
1
G(Ic2j,0)(x,t)/ dt
0
le(2j)(x) < U
=U J G 2j,0)(x~t) dt
=U Q2m2j(x),
This proves (2.1.10).
We next turn to prove (2.1.11). Due to (2.3.9), it
is enough to prove (2.1.11) for j = 1. From (2.2.14), it
follows that
(2.3.11) G 1,0) (x,t) = IJ y G (y~t) dy
+ I (y 1) Gm1(ylt) dy.
Therefore
(2.3.12) m,)xtjd
1 x
< r I y Gm1(yit)l dy dt
0 0
1 1
+ I I (1y) jm (yt dy dt.
O x
Recalling (2.2.21)
Q2m2(y) = (1)m1 I Gm1(y,t)dt ,
m = 2, 3,...
and the fact that in the unit square 0 < x < 1, O < t < 1,
(1)m Gm1(yrt) > 0
we can assert that
I1 Gm1(ylt) dt.
O
(2.3.13) Q2m2(y) =
On changing the order of integration in (2.3.12) and
making use of (2.3.13), we obtain
1
/ G ,0)(x,t)( dt &
0
x
I y Q2m2(y) dy
0
(2.3.14)
+ I (1y) Q2m2(y) dy
=: X2m2(x).
Using (2.2.20) we note that
(2.3.15) XZm2(x) = Y Q2m
On integrating by parts, we have
'(y)dy
(1 y)Q2m'(y)dy.
x
I 2m'(y) dy
0
x
(2.3.16) X2m2(x) = y Q2m 7y) i
0
1 1
Q2m'(y) (1y)l + J Q2m'(y) dy
x x
=x Q2m'(x) + (1x) Q~m'(x) + 2Q2m(x)
= (1 2x) Q2m'(x) + 2Q2m(x).
Also
(2.3.17) X 2m2 '(x) = (12x) Q2m'1(x).
Since Q~m2 vanishes only at x = 0 and x = 1, it follows
that the critical point at X2m2(x) inside [0,1] is only
at x = 1/2. Also we note that X2m2(1) = X2m2(0).
Further
(2.3.18) X2m2(1) X2m2(1/2)
= (2x1) Q2m2(y) dy > 0
1/2
Thus we conclude that X2m2(x) has an absolute maximum at
x = 0 and x = 1. Therefore, from (2.3.2), (2.3.14), and
(2.3.11), it follows that
(2.3.19) le'(x) < U G(0)xt dt
(U X2m2(x)
( U (12x) Q2m'(x) + 2Q2m(x)
I U X 2m2 (1).
On using (2.3.15) it follows that
le'(x)l IU X2m2(1) = U Q2m'(1) = U Q2m'(0),
which proves (2.1.12).
That (2.1.11) and (2.1.12) are best possible follows
from the Peano theorem, or more simply, by choosing u(x)=
Q2m(x), the Euler polynomial defined by (2.1.7). In view
of (2.2.20), we have U =: max0
use of (2.2.20) and the definition of L2m1[u,x] show that
L2m1 Q2m'x] is identically zero. Our choice of u(x) then
gives pointwise equality in (2.1.11). Similarly it can be
shown that (211)is also pointwise best possible. This
proves the theorem.
It is perhaps worth remarking that any exact
evaluation of the integral of the absolute value of a
Peano kernel results in an exact error bound (see Sard
[1963] or Stroud [1974]). Generally error bounds
resulting from integration of a Peano kernel under the
assumption that u(x) 6 Cka,lb] also hold for u having
piecewise continuous kth derivative on [a,b], and even for
u having (k1)st derivative absolutely continuous on
[a,b]. In the case given here we can thus expand the
class of functions for which the error bounds of Theorem
2.1 hold and hence are best possible.
As Theorem 2.1 is stated for function u(x) 2m times
continuous ly dif ferentiab le it also holds when the 2mth
derivative is merely piecewise continuous on [0,1].
Moreover the theorem holds even for the case that u(x) has
its (2m1)st derivative absolutely continuous. In this
last case U, instead of being the max of the 2mth deriva
tive on [0,1], becomes the "L infinity" norm of the gener
alized 2mth derivative. In the following chapters the
classes of functions k times continuously differentiable,
the class of functions having piecewise continuous kth
derivative and the class having k1st derivative absolute
ly continuous may be treated as being interchangeable.
CHAPTER THREE
MORE POLYNOMIAL ERROR BOUNDS
Introduction and Statement of Theorems
Let u e C(2m+2) [0,h] be given. It follows from a
result of Schoenberg [1966] that there exists a unique
polynomial w2m+11u,x} of degree ( 2m+1 satisfying
(3..1) w2m+ 11u,01 = u(0) ,w2m+1 [u,h] = u(h)
w2m+1 2)[u,0] = u(P (0) ,
w2m+1 2)[ulh] = u(P [u,h] ,
p = 2, 3, .. m + 1.
Theorems 3.1 and 3.2 will give bounds on u j(x)
w nl2m+1(x) for the cases m = 2 and m = 3 of polynomials
w2m+1 satisfying (..)
The polynomial w2m+1[u,x] can be expressed in
relation to the Hermite polynomial v2ml[1[ux]. To
illustrate the relation between wm+ and v~m1 let h = 1
and let v2m1[g'x] be the Hermite polynomial of degree at
most 2m 1 matching g =: u"' and its first m 1
derivatives at 0 and 1. We can represent v2m1[g,x] as
(3.1.2) v2m1[g,x] = AO(x)g(0) + BO(x)g(1)
+ A1(x)g'(0) + Bl(x)g'(1)
+ A2(x)g" (0) + B2(x)g" (1)
+ Am1(x)g[l(m)(0 + Bm1(x]g(m1)(1)
where Ai(x) and Bi(x), i = 0, 1,., m 1 are
polynomials of degree 2m 1 or less satisfying
(3.1.3) Ai ((0) 6ij A ()(1) = 0
jj =o 0, 1, . ,
Bi (0 i (1 = i '
Define for i = 0, 1, ., m 1
(3.1.4) C (x) = I G1(x, t)A (t)dt ,
D (x) = I_ Gl(x't)Bi(t)dt ,
From (314,(3.1.3) and (2.2.5)(2.2.8), it follows
that for i = 0, 1, ., m 1
(3.1.5) Ci (0)) = ij2 i 1
Di~j (0) =0 Dij )(1) =6i(j2)'
j = 2, 3, ., m + 1
where
(3.1.6) Ci(0) = Di(0) = Ci(1) = D (1) = 0 ,
and Ci, Di are polynomials of degree 2m 1 or less.
For a given u C (2m) 10,1] we can use (3.1.5) and
(3.1.6) to give w2m+1[u'x] in the form
(3.1.7) w2m+1[u'x] = u(0) (1x) + u(1) x
+ u" (0)CO(x) + u" (1)DO(x)
+ u(3)(0)C1(x) + u(3)(1)D (x)
+ u(m+1)(0)Cm1(x) + u(m+1)(1)Dm1(x)
For m = 2 and m = 3, we give (3.1.7) explicitly. For
m = 2, if u e C(6)[0,1],then the unique quintic w5[u,x]
matching u and its second and third derivatives at 0 and 1
is given by
(3.1.8) w51u,x] = (1x) u(0) + x u(1)
+ u" (0) [7x/20 + x2/2 x4/4 + x5/10]
+ u" (1) [3x/20 + x4/4 x5/10]
+ u" '(0) [x/20 + x3/6 x4/6 + x5/20]
+ u"'(1)[x/30 x4/12 + x5/20].
For u 6 C 8)[0,11, the unique polynomial w7[u,x] of
degree 1 7, matching u and its second, third and fourth
derivatives at 0 and 1 is given by
(3.1.9) w7[u,x] = (1x) u(0) + x u(1)
+ u" (0) [5x/14 + x2/2 x5/2 + x6/2 x7/7]
+ u" (1) [x/7 + x5/2 x6/2 + x7/71
+ u(3) (0) [13x/210 + x3/6 3x5/10
+ 4x6/15 x7/14]
+ u(3)(1) [4x/105 x5/5 + 7x6/30 x7/14]
+ u(4)(0) [x/210 + x4/24 3x5/40
+ x6/20 x7/841
+ u(4)(1) [x/280 + x5/40 x6/30 + x7/84].
The following theorem concerns the quintic
interpolant w5*
Therem3.1 Let u e C6[O,11 and let w5[u,x] satisfy
(3.110) w5 P)1u,0] = u 9)(0) ,
w5 2)[u,1] = u 2)(1) ,p = 0, 2, 3.
Denote
(3.1.11) e(x) = u(x) w51u,x]
and
(3.112)U = max0
Then for 0
pointwise bounds hold:
(3.1.13) le 2)(x) (< U f ~(x)
where
f0,0(x) = [ x3(1x)3 + x2(1x)2/2 + x(1x)/2 ] / 61
f0,1(x) = [ 1/60 x3(1x)3/3 ] / 41,
f0,2(x) = [ x21x2]/4
f0,3(x) = [ x(x1)(2x1) ] / 12 O < x < 1/3
= [ 16x3 105x2 + 197x 162
+ 66/.x 13/x2 + 1/x3 ] / 96 ,
1/3 < x < 1/2
f0,4(x) = [ 485 + 42x 0x
+ 54x2 12x + 1 ]/1()
0
=[ 6(x1/2)2 + 1/2 ] / 12 1/3 < x < 2/3
f0,5(x) = (x1/2)4 + 3(x1/2)2/2 + 3/16 O < x < 1
and where f0,2 and f0,3 are extended to the whole of (0,11
by even symmetry about 1/2.
Furthermore, the functions f0,p, p = 0, 2, 3, 4, and
5 are pointwise best possible. The functions f0l,2 0,3'
f0,4 and f0,5 are those of Birkhoff and Priver [19671 for
two point cubic interpolation.
That these functions also serve as error bounds in
the present case is a consequence of the fact that
w5l[ux]' is the unique cubic matching u" and u"' at 0
and 1. In other words w5" [u,x] is the Hermite cubic
interpolation vj[g,x] where g = u". The error bounds
given by Birkhoff and Priver in terms of max0
are now expressed in terms of U = max0
g(4) is in fact u(6))
Denoting
(3114 p = a0x< ,(x)j p = 0, 1, . 5
we have
c = 11 1 c = 1 1 c = 1
646! 2 61 24 4!
c3 = 1/ c = 1 c5 =1.
9 4! 12 2
From (3.1.14) and (3.1.13) it follows that for every
u e C(6)[0,11
(3.115)max0
Remark 3.1 Note that
cp = max0
If we set u(x) = f0,0(x) then we have
e~x = 0,0(x w7 f0,0,x
=f0,0(x) and U = max0
By Remark 3.1 we see that for u(x) = f0,0(x) equality is
attained in (311)for p = 0, 1, 2, 3, 4, 5. The
constants cp are thus the smallest possible.
The next~theorem gives error bounds for w7, analogous
to the error bounds for w5 given in Theorem 3.1.
Theorem 3.2 Let u C C 8)[0,1], and let w7 [u,x] be a
polynomial of degree 7 or less satisfying
(3.116)w7 9)[u,0] = u (0) ,
w7 2)[u,11 = u 9)(1) p = 0, 2, 3, 4.
Denote
(3.117)e(x) = u(x) w7[u,x]
and denote
(3.118)U = max0
Then, for 0 < x < 1 and 0 < p < 7, the following
pointwise bounds hold:
(3.119) e 2)(x)l < U f l(x)
where
fl,(x) = [: x4(1x)4 + (2/5)x3(1x)3
+ x2(1x)2/5 + x(1x)/5 ] / 81
fl,1(x) = (1/5)(1/81) (1/4)(1/61)x4 (1x)4
fl,(x) = x3(1x)3/6!
fl,3(x) =X x(x1)2(12x)/240 O
=x2(x1)2(12x)/240
+ 4(x1)2 10T x2
+ 2T(15x2+2x+1) + 5x(5x2) ] / 120,
2/5 < x <1/2
where
T = [ (3x1)(5x+1) + (x1)(15x2+6x+1)1/2 / 12
f (X) = x(1x)(525+)1
1,4
=x(1x) (5x25x+1)/120
+ T14 [ 2T12(2x33x2+x)
+ 12T1(5x3+8x2x/
+(10x318x2+9x1) ]/ 12,
for (4J )/10 < x (3/T)/6
where
T1 = [ 15x2 9x (x1) (3x(45x))12/6(x),
1,(x) = x(x1)(5x25x+1)/120
+ W4x ( 10W2(2x23x+1)
+ 4W(15x221x+6)
+ 5(10x212x+3)
+ 5(10x212x+3) ]/60 ,
for (3i'j)/6 < x < (6/ ~)/10
and where
W= [ 3(1x)(5x2) + x(3(1x)(5x1))1/2
6(x1) (2x1)
fl,4(x) = x(x1)(5x2 5x+1)/120 ,
fl,5(x) = (2x1) (10x210x+1)/120
+W4 [ 2W2626x+1)
+ 24W1(15x214x+2)
+ 30(10x28x+1) ] / 120,
O < x < (4fl)/10
where
W1 = [ 15x2 14x 2 x(3x(45x))1/
12x2 12x + 2
=(2x1)(10x210x+1)/120 ,
(4J7)/10 ( x ( (6J6)/10
=(2x1) (10x210x+1)/120
T4 [ 20T22(6x26x+1)
+ 24T2(15x2+16x3)
+ 30(10x212x+3) ] / 120,
(6 6)/10 < x < 1/2
where
T2 [15216+3 (x1)(15x2+18x3)1/
12x2 12x + 2
fl,6(x) = [ 15x2 + 5x 1 ]/1
W24 2(x1/2)
+ W2(15x7)/5 + 5x/2 1 ]
O < x < 2/5
where
W2=[1x+7(15x+6x+1)1/ ] / (12x6)
fl,6x) (x1/22/2 + 1/40 2/5 < x < 1/2
fl,7(x) = 2(x1/2)6 5(x1/2)4/2
+ 15(x1/2)2/8 + 5/32 ,
O
and where fl, ,4 1,5 and fl,6 are extended to
(1/2,1] by symmetry about x = 1/2. Furthermore, each of
the functions fl~ where p = 0, 2, 3, 7 is pointwise
exact.
Setting p = max0
(3.1.20) dO a l lf
d 128 ,! d = 5
2 61 26' 3 30,000
d6 0 d7 Z
From (3.1.19) and (3.1.20) it follows that for
(3.1.21) max0
Remark 3.2 Analogously to Remark 3.1, note that
(3.1.22) max0
On setting u = fl,0 (x) it follows from (3.1.22) that
(3.1.21) is exact for each p.
The following would seem to a natural generalization
of the Theorems 3.1 and 3.2.
Conjecture 3.3 Let u e C(2m+2)[0,1] and let w2m+11u,x] be
the polynomial of degree at most 2m + 1 matching u and
its 2nd, 3rd, . (m+1)st derivatives at 0 and 1.
Denote
(3.1.23) e(x) = u(x) w2m+11u,x]
and
(3.1.24) U = max0
Then for p = 0, 1, 2, we have
(3.1.25) le 2)(x)( (U fm1,p~x)
where
fm1,0(x)= (1) ( ) [xm+2+ix] } / (2m)!
i=0 [(m+i+2)(m+i+1)]
f1(x) =1 xm+(1x)m+ ) / (2m)! '
(2m+2) (2m+l) m+1
fm,(x) = ( xm(1x)m 3 / (2m)!.
Furthermore (3.1.25) is pointwise exact, p = 0 and 2.
Analogously to Remarks 3.1 and 3.2, it may be that
for every u e C(2m+2)[0,1] and p = 0, 1, ., 2m+1
(3.1.26) max0
If Equation (3.1.26) holds then it is best possible as can
be verified by choosing u = fm1,0 and noting that then
e~x) is the same as fm1,0(x). For p = 0, 1, 2,
max0<< m (x)J = max0
Hence if (3.1.25) holds then (3.1.26) is true for
p = 0, 1, 2. As
fm (2) (x) = [xm(1x)m)/(2m)! ,
the conjecture of (3.1.26) is related to the following
conjecture.
Conjecture 3.4 Let u 8 C(2m)[0,1] and let v2m1 be the
Hermite polynomial of degree at most 2m1 matching u and
its first m1 derivatives at 0 and 1. Denote
U = max0
and
e(x) = v2m1[u'x] u(x).
Then
max0
i U max0
p = 0, 1, 2, . 2m1.
The results of Birkhoff and Priver demonstrate Conjecture
3.4 for the cases m = 2 and m = 3. Recent work of Bojanov
and Varma indicates that Conjecture 3.4 is in fact true.
The next theorem will concern an interpolatory
polynomial which enjoys a similar property to that of
the above conjectures. Let u C C(4 [0,1]. Define
k3 [u,x] by
(3.1.27) k31urx] = u(0) (1x)(12x)2 + u(1/2) 4x(1x)
+ u(1) x(12x)2 + u'(1/2) 2x(1x)(2x1).
Then k3[u,x] is the unique polynomial of degree 3 or less
satisfying
(3.1.28) k3[u,x] = u(0) k31u,1] = u(1)
k3[u,1/2] = u(1/2) k3'[u,1/2] = u'(1/2).
Theorem 3.3. Let u 6 C(4)1[0,1]. Denote
e(x) = k3[u,x] u(x) ,
U = max0
Then for p = 0, 1, 2, 3, we have
(3.1.29) le(P(x)l < ap U
where
a0 = 1 / (28 4!) al = 1 / (22 4!)
a2 = (5/2) (14) = 1/2.
That the ap are the best possible can be verified by
choosing
u(x) = [ x(1x)(12x)2 ] / (22 4!).
Due to the similarity between the proof of Theorem
3.3 and several other proofs in the following chapters, it
would be redundant to prove it here.
Proof of Theorem 3.1
Let u e C(6) [0,1]. Then
(3.2.1) w51u,x} = (1x) u(0) + x u(1)
+ u" (0) [7x/20 + x2/2 x4/4 + x5/10]
+ u" (1) [3x/20 + x4/4 x5/10]
+ u" '(0) [x/20 + x3/6 x4/6 + x5/20]
+ u" '(1) [x/30 x4/12 + x5/20]
is the only polynomial of degree L 5 satisfying
(3.2.2) w5 2)[u,0] = u 2)(0) ,
w5 P) [u,1] = u 2) (1) ,p = 0, 2, 3.
Define
(3.2.3) e(x) = u(x) w5[u,x].
Then
(3.2.4) e 2)(0) = 0, e(P (1) = 0, p = 0, 2, 3,
and
(3.2.5) e(6)(x) = Q(x) =: u(6)(x).
In other words, e(x) is the unique solution of the
differential equation (3.2.5) with boundary conditions
(3.2.4). We can rephrase (3.2.4) and (3.2.5) as
(3.2.6) d~e = y(x) ,
dx2
e(0) = 0, e(1) = 0
and
(3.2.7) d'y = Q(x)
dx4
y(0) = y(1) = y'(0) = y'(1) = 0
From (3.2.6) and (3.2.2)(3.2.6), it follows that
1
(3.2.8) e(x) = G1(xz) y(z) dz
where
z(x1) O < z < x < 1
Gl(x'z)=
x(z1) O < x < z < 1
is the Peano kernel for linear interpolation used in the
proof of Theorem 2.1.
Similarly, from Birkhoff and Priver (or by applica
tion of the Peano theorem), we have
(3.2.9) y(z) = I G4 (z,t) Q(t) dt ,
where
(3t22t3 g3 + 3(t2)t2z2
+ 3t z t3
(3t22t31)z3 + 3(t1)2tz2
,t < z
6G4(z,t) =
t >
for 0 < t < 1, O <
Combining (3.2.8) and (3.2.9) we have
z < 1
= G1(xZ)
= Gl(x,2)
j_ G4(z't)
I_ G4(2,t)
(3.2.10) e(x)
Q(t) dt dz
u(6) (t) dt dz
u(6) (t) dt dz
dz u(6)(t) dt
1 1
= I G1(x'Z) G4(z't)
0 0
1 1
= /G1(x~,z G4(z~t)
0 0
1
= G(x,t)
0
u(6)(t) dt
where
(3.211)G(x,t) = I G1(x,2) G4(z,t) dz.
From (3.2.11) and (2.2.5)(2.2.8), it follows that
(3.2.12) G(2,0)(x,t) = G4(x,t)
and
(3.2.13) G(p+2,0)(x,t) = Gq(p,0)(x,t) p = 0, 1, 2, 3.
Also, as
G4(z't) I 0 O < z < 1 O < t < 1
G1(x'z) I 0 O < x < 1 O < z < 1
it follows that
0 < x < 1 O < t < 1
G(x,t) > 0
From (3.2.10) and G(x,t) 0 we have
le(x)j I I 1G(x,t) dt max0
0
(3.2.14)
In fact,
1
IJI Gq(x,t)dt dx dx + ax + b
0
(3.2.15) I G(x,t)dt =
0
where a and b are chosen to satisfy
(3.2.16) ii G(0,t)dt = G(1,t)dt = 0
O 0
We know from Birkhoff and Priver (or Hermite) that
1 G4(x,t)dt = [x2(1x)2]/4).
O/4!
Then
1
I G (x,t)dt dx dx = 1
0 61
( 5x4 + 3x5 ,6)
2
and to satisfy (3.2.16), we have a and b of (3.2.15) as
a =1/2 ,
61
b = 0.
Rearranging, we have
1
SG(x,t)dt =[ 5x4/2 +35 6+x/ ]/6
(3.2.17)
=[x3(1x)3 + (1/2)x2(1x)2 + (1/2)x(1x) ] / 6!
=f0,0(x) .
Combining (3.2.14) and (3.2.17), we have the result of the
theorem for p = 0.
From 3.2.10, we have
e 9)(x~l CI 1 G 9)(x,t)l dt max0
O
(3.2.18)
From 3.2.11, we have
x 1
(3.2.19) G(1,0)(x,t) = I y G4(y,t)dy + I (y1)G4(ylt~dy.
Therefore as G (y,t) I 0
0
x
lG(1,0) (x,t) 1( y G (y~t) dy
0
(3.2.20)
+ / (1y) IG4(y,t) Idy.
As before, we have
i jG4(yit)ldt= y2(1y)2 / 41
Thus
(3.2.21) / G(1,0)(x~t) dt < IJ Jy 1G (y~t)l dy at~
0 0 0
1 1
+ j I (1y) IG4(ylt)l dy dt
0 x
x 1
= y I IG4(yit) dt dy
0 0
1 1
+ J (1y) I 34(yit)l dt dy
x 0
= J [ y3(1y)2 ] / 4!dy
+ fl [(y)3 2 / 4! dy
=[ 1/60 (x3(1x)3)/3 ] / 4!
=f0,1(x)
which achieves its maximum value of 1/1440 for x = 0 or
x = 1. We note also that
1/1440 = 1/(2 6!) = C1
=max0
Combining (3.18) and (3.13), we have
(3.2.22) le 2)(x) <
II G4(p2,0) (xt) dt max0
for p = 2, 3, 4, 5.
As this inequality is precisely that used by Birkhoff
and Priver to derive the functions f02 0,3 0, and
59
f05 the theorem follows for p = 2, 3, 4, 5. The proof
of Theorem 3.2 is very similar and hence omitted.
CHAPTER FOUR
A QUARTIC SPLINE
Introduction and Statement of Theorems
Among the many beautiful properties of the complete
cubic spline is the fact that for a given partition and
function values, the cubic spline is obtained by solving a
tridiagonal ly dominant system of equations.
Unfortunately, when one uses higher order complete splines
the bandwidth grows. In fact, for a 2m times continuous
spline of order 2m+1, the bandwidth of the system of
equations is 2m+1. Furthermore the diagonal becomes less
dominant as k increases.
It is natural then, to increase the order of the
spline but preserve bandwidth. Ideally we would hope to
increase the diagonal dominance and order of convergence.
In this chapter we introduce a quartic C(2) spline which
gives O(h5) rate approximation to a C(5) function. The
quartics are obtained by the solution of a tridiagonally
dominant system. As desired, it is more diagonally
dominant than the system associated with the complete
cubic spline.
The main result of this chapter will be to give an
exact error bound for the quartic spline discussed here.
We first give the definition.
60
Let f be a realvalued function defined on [a,b).
Choose a partition (xi ~=0suhta
a = x0 < x1 < . < xk = b .
Let zi = (xi1 + xi)/2, be the midpoint of [xi1, xi) for
i = 1, 2, .. k and for these i set hi1 = xi xi1'
Definition 4.1 Given the function f and the partition
{xi=,0', w define a quartic spline s(x) such that
(4..1) s~x 6 2[a,b] FT P4[xi1, xi ] i = 1, 2, k ;
(where P4 [xi1, xi] denotes the functions which are
quartics when restricted to [xi1, x ])
(4.1.2) s(xi) = f(xi) for i = 0, 1, ., k
s(zi) = f(zi) for i = 1, 2, ., k ;
and
(4.1.3) s'(a) = f'(a) and s'(lb) = f'(:b).
Lemma 4.1 Let f be a realvalued function defined on
[a, b] an e x 0b a partition of [a,b]. A
quartic spline s satisfies Equations (4.1.1) and (4.1.2)
if and only if s satisfies the tridiagonal system of
equations for i = 1, 2, ., k 1
(4.1.4)
hi s'(xi1) + 4(hi + hi1) s'(xi) hi1 s'(xi+1)
+ 16 [(hi1/hi) f~zi+1) hih) i
5 [(hi1/hi) f(xi+1) 'hi/hi1) f(xi1)]
where hi = xi+1 xi'
We will give the proof later.
Assuming from (4.1.2) that f (xi), i = 0, 1
and flzi), i = 1, 2, . k are known, then (4.1.4) is a
system of k 1 equations in the unknown variables
s'(i),i =1, .. ,k.If we impose the conditions of
(4.1.3) that s' (x0) = f' (a) and s' (xk) = f' (b) are given,
we have k1 unknowns and the k 1 diagonally dominant
equations (4.1.4). Lemma 4.1 thus assures us that s' (xi)
can be uniquely determined for given conditions (4.1.1)
(4.1.3). As will be shown in the proof of the lemma, there
is, on any given subinterval [xi, xi+11, a unique quartic
si(x) satisfying the five conditions
(4.1.5) si(xi) = f(xi) si(zi+1) = f(zi+1) '
si(xi+1) = f(xi+1) s'i(xi) = s'(xi)
s'i(xi+1) = s'(xi+1 *
Equations (4.1.4) are derived by imposing the conditions
that
si"((xi+) = i1(i)for i = 1, 2,.., k 1.
For i = 1, 2, ., k, si(x) is thus the restriction of
the spline s to lxi, xi+l]*
Summarizing, unique solution of (4.1.4) implies that
s(x) is uniquely defined on each partition subinterval
[xi, xi+1], i = 0, 1, ., k1, which is to say, on all
of [a, b]. We have shown
Corollary 4.1 The quartic spline of Definition 4.1 is, for
a given partition {xi=,0 and function f, unique.
We now make the comparisons with the complete cubic
spline more explicit. The system of equations
corresponding to (4.1.4) for the complete cubic spline has
lefthand side
his'(xi1) + 2(hi + hi1) s'(xi) + hi1s'(xi+1)
In comparison (4.1.4) is twice as diagonally dominant.
To interpolate the 2k + 1 function values f(xi) and
f(zi) using our C(2) quartic required solving the
tridiagonal system of k 1 equations (4.1.4) As the
cubic spline must match derivative and second derivative
values at each interior function value, interpolation of
the same 2k + 1 function values by the C(2) cubic spline
would entail solution of a system of 2k 1 equations. In
other words, the matrix equation to be solved for the
quartic is only half as large as that required for the
cubic.
We can now state the main theorem of this chapter.
Given a partition {x @=,0of[,bdnt
h = max0
For each x in [a,b], there exists i such that
0 < i < k 1 and xi < x
Theorem 4.1 Let f e C(5)[a,b] and let {xi }kj be a
partition of [a,b). Let s(x) be the twice continuously
differentiable spline corresponding to f and {xi =,0'
where s satisfies (4.1.1) (4.1.3) Then
(4.1.6) (f(x) s(x)l < Ic(t)l h5 maxa(x(b g(5)(x)) / 5!
where
c(t) = [3t2(12t)(1t)2 + t(12t)(1t)] / 6.
Define
C0 = max0
=(1 1 ) (1/5 + 2/15 ) / 6.
It follows that
(4.1.7) f(x) s(x)j ( co h5 ma~xaxb lf(5) (x) / 51
Furthermore, neither Ic(t) nor co can be improved, as we
can show by letting f = x5/5! and letting k become
arbitrarily large for an equally spaced partition. An
approximate decimal expression for co is .0244482 and
c /5! is approximately .000203818.
We will also show
(4.1.8) Jf'(xi) s'(xi)
& 4 maax f (5)(x)l / 6!
and that this estimate is exact.
Related to Theorem 4.1 is the following conjecture.
Conjecture 4.1 Let f e C(5)[a,b] and let {xi }k2 be a
partition of [a,b). Let s(x) be the twice continuously
differentiable spline corresponding to f and {xi =,0'
where s satisfies (4.1.1)(4.1.3). Then
(4.1.9) (f'(x) s'(x)( A4axxb 5)) /6.
If Conjecture 4.1 holds, then the constant 1/6! can not be
improved. This conjecture has been verified numerically.
Remark 4.1 Given f e C5[a,b] and a partition {x}k, of
[a,b], let s be the quartic C(2) spline satisfying
(4.1.1) (4.1.3) Then the supremum norm  f~i s~i
is of order h5 maxaxb(5x),i=01,2
Theorem 4.1 demonstrates that the quartic C(2) spline
gives the best possible order of approximation to
functions from the smooth class C(5). We next discuss
interpolation to the much less smooth class of functions
which are merely continuous on [a,b]. As f'(a) and f'(lo)
are not necessarily defined, we consider the quartic C(2)
spline satisfying (.1)and (4.1.2) with boundary
conditions
(4.110)s'(a) = s'(:b) = 0
Denote w(f,h) =: sup xy ghlf(x) f(y) .
Theorem 4.2 Let f e Cla,b]. If {x }k= isthpaiio
of equally spaced knots, then for xi < x < zi+1 = (xi +
xi+1)/2 and t = (x xi)/hi, i = 0, 1, k 1, we
have
(4.1.11) f(x) s(x)l L c(t) w(f,h) ; 0 < t < 1/2
and for zi+1 < x < xi+1, or 1/2 < t < 1
(4.112) f(x) s(x)l L c(1t) w(f,h)
where c(t) = (1 + (13/3)t 3t2 (58/3)t3 + 16t43
Note that max0
1.6572.
The bound of the preceding theorem is only valid for
equally spaced knots. For arbitrary partitions we can not
give a bound of this same form. However, if
m =: max0
we have the following theorem.
Theorem 4.1.3 Let f(x) e C[a,b], and let s be the C(2)
quartic spline satisfying (1.)(412,and (..)
Then for xi < x < zi+1, and i = 0, 1, 2, ..,k 1,
(i.e., for 0 < t <1/2 with t = (x x )/h )
(4.113) f(x) s(x) < c (t) w(f,h)
and for zi+1 < x < xi+1, i'e. 1/2 < t < 1,
(4.1.14)f(x) s(x) < c (1t) w(f,h)
where
cl(t) = [1 + 10t2 28t3 + 16t4]
+ (8/3)[m2 + m] [t(12t)(1t)]
Theorems 4.2 and 4.3 indicate that for suitable
partitions the quartic C2 spline can provide acceptable
approximations to functions which are merely continuous on
[a,1b].
Proof of Lemma 4.1
We first give an expression for the unique quartic
matching function and derivative values at endpoints and
function values at the midpoint. Specifically, let f be a
realvalued function defined on [0,1], and differentiable
at 0 and 1. Let
(4.2.1)
P (x) = 1 11x2 + 18x3 8x4 =(2x1)(1+4x)
P2(x) = 16x2 32x3 + 16x4 = 16x2(1x)2
P3(x) = 5x2 + 14x3 8x4=(12x)x2[1+4(1x)]
P4(x) = x 4x2 + 5x3 2x4 x(1 2x) (1x)2
P5(x) = x2 3x3 + 2x4 = x 2(12)1x
Then
(4.2.2) L[f,x] = Pl(x)f(0) + P2(x)f(1/2) + P3(x)f(1)
+ P4(x)f'(0) + P5(x)f'(1)
is the unique quartic satisfying
(4.2.3) L[f,01 = f(0) L[f,1/21 = f(1/2),
L[f,11 = f(1) L'[f,0j = f'(0) L'[f,1] = f'(1).
L is a linear functional and a projection. If f is a
polynomial of degree four or less, then L[f,x] = f(x). In
the future calculations we will need the following facts
about the quartics Pi'
(4 2. ) P "() =22 Pl (1 = 0
P2 (0) =3P2" (1) = 32
P3 (0) =0P"1)= 22
P4" (0) =8P4" (1) = 2
P5 (0) = 5" (10) = 8.
Let zi+1 = (xi + xi+1)/2. On the interval [xi, xi+1 '
the unique quartic Li[f,x] interpolating f(xi), f'(xi '
f(zi+1), f(xi+1), and f'(xi+1) can be expressed in terms
of Pi. In fact, let t = (x xi)/hi where hi = xi+1 xi'
Then
(4.2.5) Li[f,x] = f(xi) 1l(t) + f(zi+1) P2(
+ f(xi+1) 3(t) + hi f'(xi) 4q(t)
+ hi f'(xi+1) 5(t).
Let s be the quartic spline of Definition 4.1 corres
ponding to f and the given partition. Then the restric
tion si(x) of s to [xi,xi+1] is a quartic. Hence
Lils' x] = s (x). Using' the facts that s(xi) = f(xi '
s(Zi+1) = f(zi+1), and f(xi+1) = s(xi+1) in (4.2.5) we
have
(4.2.6) si(x) = f(x ) Pl(t) + f(zi+1) P2(
+ f(xi+1) P3(t) + hi s'(xi) P4(
+ hi s'(xi+1) 5g(t) t = (xxi)/hi
In order that s be twice continuously differentiable,
we must satisfy
(4.2.7) si" (xi+) = si1" (xi
where si is the restriction of s to lxi, xi+1] and si1 is
the restriction of s to lx ~,x ]. Differentiating
(4.2.6) twice we have
(4.2.8) s" (x +) = _1 f(xi P1" (0) + f(zi+1) 2" (0)
+ f(xi+1) P3" (0) + hi s'(xi) 4" 1(0)
+ hi s'(xi+1 5" 1(0) 3.
Similarly, from rewriting (4.2.6) for the interval
[xi1, xi], we have
(4.2.9) s" (xi_) __1 ( f(xi1 P1" (1) + f(zi P2" (1)
hi1
+ f(xi) P3"(1)
+ hi1 s'(xi1) 4"l(1)
+ hi1 s'(xi) 5" l(1) 3.
Setting s"(xi+) = s"(xi) by equating (4.2.8) and
(4.2.9) and using Pi"(0) and P"(1) from (4.2.4), we
have
(4.2.10) {22 f(x ) + 32 f(zi+1) 10 f(xi+1)
8 hi s'(xi) + 2 hi s'(xi+1) 3 / hi2
= 10 f(xi1) + 32 f(z ) 22 f(xi)
2 hi1 s'(xi1) + 8 hi1 s'(xi) } / hi12
Factoring two, multiplying by hihi1, and putting the
known function values on the right hand side, we have
(4.211)his'(xi1) + 4(hi+hi1) s'(x ) hi1s'(xi+1)
=11 [(hi1h ih) ~i
+ 16 [(h1/ifzi1 h/i1fzi
5 [(h1/ifxi1 hih1)fx1
which is the desired system of equations (414. Having
established Lemma 4.1, we next turn to a proof of Theorem
4.1.
Proof of Theorem 4.1
Our method of proof is to establish a pointwise
bound. As in the proof of Lemma 4.1, let L (f,x] be the
unique quartic agreeing with f(xi), f(xi+1), f(zi+1 '
f'(x ), and f'(xi+1), and let s be the twice continuous
quartic spline corresponding to f and Equations (.1)to
(4.1.3) on the partition (xi =,0. Then for xi < x < xi+1'
we have
(4.3.1) If(x) s(x)l < If(x) L [f,x]j
+ (L (f,x] s(x).
Assume that f e C(5)1a,b]. By a proof attributed to
Cauchy, we know that
(4.3.2) If(x) L (f,x]l < (hi5/5!) lt2(1/2t)(1t)21 U
where t = (xxi)/hi and U is the maximum of f(5)(x) on
[x ,xi+1]. Equation (4.1.9) gives a pointwise bound for
jf(x) L [f,x].
Let i be arbitrary and xi < x < xi+1. We next turn
our attention to deriving a similar bound for
 L [f ,x] s (x) =  L [f ,x] s (x)  Subtracting (4.2.6)
from (4.2.5) gives
(4.3.3) Li[f,x] si(x) = hi [f (xi) s'(xi)l 4qt
+ hi If'(xi+1) s'(xi+1~ 5gt *
Denoting
(4.3.4) e'(x ) = f'(xi) s'(xi)
then we have from (4.2.5),
(4.3.5) IL (f,x] s (K)l <
hi max(e'(xi)l e'(xi+1)l 4 5 *ll~~t l
As Pq(t) = t(12t)(1t)2 an 5t 2(12t)(1t)
are both positive for 0 < t < 1/2 and both negative for
1/2 < t < 1, IP4 ~ ~ 5~(t)l = IP4(t) + P5(t)l for
0 < t < 1. Then for xi < x < xi+1, we have
(4.3.6) IL [f,x]s(x)l <
hi maxcle'(x ) e'(xi+1) 3 t(12t)(1t).
Redefine L so that its restriction to [xi, xi+1] is Li for
each i, i = 0, 1, . ,k1. Choose i so that je'(xi)l is
maximal. We then have for all a < x < b,
(4.3.7) IL[f,x] scx)l L h le'(x ) t12t)1t
where h = max0
and where on each subinterval [x xj+1J, O < j < k 1,
we define t = (x x )/h .
The next task is to bound le'(xi)). From both sides
of (4.1.4) we subtract
hi f'(xi1) + 4(hi + hi1)f'(xi) hi1f'(xi+1) '
thereby defining a functional BO f)
(4.3.8) hie'(xi1) 4 (h +hi1) e'(:xi) + hile'(xi+1)
=hi f'(xi1) 4(hi+hi1)f'(xi) + hi1f'(xi+1)
11 [(hi1/hi] (hi/hi1)] f(xi)
+ 61hi1/hi) f(zi+1) hi/hi1) f(zi)l
5 [(hi1/hi) f(xi+1) hi/hi1) f(xi1~
=: BO n *
The linear functional BO(f) is identically equal to
zero when f is a polynomial of degree four or less, as can
be directly verified. (The arithmetic of verification is
simplest if one takes xi1 = hi1, xi = 0, xi+1 = hi and
checks the monomials 1, x, x2, 3, and x4,
We have chosen i so that Je'(xi)j attains its maximum
value. As
4(hi + hi1)e'(xi)
=BO(f) + hie'(xi1) + hile'(xi+1)'
it follows that
j4(hi + hi1) L ]BO(f)l + jhie'(xi1)
+ Jhile'(:x )
I 1BO(f) + I(hi + hi1)e'(xi)
Hence
13(hi + hi1)e'(xi)l 0 ~g
and
(4.3.9) e'(xi)l 0 ~(f) / 13(hi+hi1) *
As BO(f) is a linear functional which is zero for
polynomials of degree four or less, we can apply the Peano
theorem to get
(4.3.10) BO~f x i+Ol(xy) 4] f(5)(y) dy / 41
xi1
From (4.3.10) follows
(4.311)BO xi+1 I BO[(xy) 4]1 dy Ui/ 4!.
xi1
where Ui is the maximum of [f(5)1 on [xi1 xi+1 *
For xi1 < y < x ,l B01(xy) 4] takes the form
(4.3.12) Bgl(xy) 4]
16(hi+hi1) (xiy)+3 + 4hi1 (xi+1_ 3~
1 [hi/h) hihi1) h/il(xi *+4
+ 16 [(hi1/hi)(zi+1~7)+4_h/i1(iY+
5 (h1/i(xi+1 y)4
In order to evaluate the integral of (4.3.11) we need
to know the sign behavior of BO[(xy) 4]. We rewrite
(4.3.12) in a form which shows its symmetry about xi.
(4.3.13) BO[(xy) 4]
(hi/i1)[5(xiy) + hi11 [(xi y) hi1 3
for xi1 < y < Zi
(hi/hi1) (xi 2)[11(xi_ 2 16hi1(xiY
+ 6hi2]
for zi < y < xi
(hi1/hi) (x y)2 [11(x y)2 + 16hi(xiY
+ 6hi2]
for xi < y < Zi+1
(hi1/hi) [5(xiy) hi] [(xiy) + h1 ]3
for zi+1 < y < xi+1
As the expression (4.3.13) has factors which are at
most quadratic it is fairly easy to to determine to
determine the sign of BO[(xz) 4]. In fact, BO[(xz) 4]
is nonnegative for i1 I y &xi+1. Evaluation of (..1
is then straightforward. The term by term integration of
(4.3.13) gives
(4..1) xi+1BO[(xy) 4] dy = hih1i1 h 3]/10.
xi1
From Equation (4.3.11) we conclude that
(4.3.15) IBO(f)l ( Ui hihi1[hi1 + hi3 25)
From (4.3.9) it is then evident that
(4.3.16) le'(x )l <
U hhi1[i13 + h3] /[(6!)(hi+hi1~
for = 2 1. As
[h3 +h3] / hih1 mxh2, hi123
and as
Gi < U,
it follows that
(4.3.17) max le'(x ) I max(hi4, hi14} U/(6!).
This is the desired bound on I e'(xi) *
Applying it in (4.3.7) we have
(4.3.18) IL[f,x] s(x)l L h5 t(12t)(1t)l U/ (61).
From (4.3.2) follows
(4.3.19) f(x) L[f,x] ( h5 lt2(t1/2)(1t)2 U/ 5!
where L restricted to [xi, xi+11 is defined as L [f,x] and
where h is the maximum of hi'
We can now combine the bounds on If(x) L[f,x] and
L[f,x] s(x~l. From (4.3.19) and (4.3.18), we have
(4.3.20) If(x) s(x)j ( h5 Ic(t)l U / 5!
where
(c(t)[ = 13t2(12t)((1t)2( + It(12t)(1t)l / 6
= l3t2(12t)(1t)2 + t(12t)(1t)l / 6
and
c(t) = [3t(1t) + 11 [t(12t) (1t)] / 6*
Then
(4.3.21) co = max0
To verify (4.3.21), note that
(4.3.22) 6c'(t) = 30t2(t1)2 + 1
=30 [(t 1/2) + 1/2]2 [(t 1/2) 1/2]2 + 1
=30 [(t 1/2)2 1/4]2 .
For 0 < t < 1, the roots of c'(t) are
(4.3.23) t = 1/2 + /1/4 1//~i .
Evaluating c(t) at the roots of o'(t), we get
(4.3.24) CO = ( J1/4 1//TEi ) (1/5 + 2//TE ) .
We have shown the socalled "direct" part of the
proof, that Equation (4.1.7) holds for co. It remains to
be shown that the theorem holds for no smaller c .
In fact, given c < co, we can produce a function f
and a partition {x 3 =0 of 11, 1] such that
(4.3.25) max1
c h5 max1
Often, when polynomial interpolation of degree n is
considered, the worst error is attained by a polynomial of
degree n + 1. As s is a quartic spline, it is natural to
try f(x) = x5/5! as a possible worst function. A particu
larly pleasant feature of the trial worst function f is
that it has fifth derivative identically equal to one.
For xi < x < xi+1, we have by the Cauchy formula
(4.3.26) x5/5! L [x5/5!,x]
=h5[t2(t1/2)(t1)2] / 51
Furthermore, for equally spaced knots xi1, xi, xi+1, we
can calculate
(4.3.27) BO(x5/5!) = hi/5.
If e' (xi1) = e' (xi) = e' (xi+1)r ,we have from (4.3.8)
(4.3.28) e'(xi) = BO(x5) / h4/6.
Equation (4.3.3) then becomes for f(x) = x5/5!
(4.3.29) L (f,x] s(x) = hi hi4 l94t 5 P(t)}/6!
= i [t(2t1)(1t)]/61
Combining Equations (4.3.26) and (4.3.29), we have,
for xi < x < xi+1'
(4.3.30) f(x) s(x) = hi5 {~/)1t/
+ t2(t1/2)((1t)2} / 5!
As (4.3.30) gives, after taking its absolute value,
precisely our pointwise bound Ic(t)j of (4.3.20), we will
have attained c provided only that hi = h, and as men
tioned above,
(4.3.31) e'(xi) = e'(xi+1) = e'(xi1) = h4/61
In order that hi = h, we take the knots to be equally
spaced. Attaining (4.3.31) is not so easy. In fact it is
attained only in the limit. The difficulty is the boundary
conditions e' (x0) = e' (xk) = 0. We can show, however,
that as one moves many subintervals away from the
boundaries, e'(xi) goes to h4/61.
Explicitly, let {x ) =0 be the partition dividing
[1,1] into k equal subintervals; in this case, h = hi
2/k For i = 1, 2, .., k 1, and f = x5/5!, we have
BO(f) defined on [xi1, xi+11 and
(4.3.32) BO(f)/h =h/5! = e'(xi1) 8e'(x ) + e'(xi+1)
We wish to apply (4.3.32) inductively to move away
from the end conditions e'(1) = e'(1) = 0. In order to
do so we must establish that e'(x ) < 0 for 0 < i < k. We
reason by contradiction.
Let 1 < i < k 1. Suppose e'(x ) > 0. Then
e'(xi 1) + '(xi 1
e'(xi1) 8 e'(x ) + e'(xi+1)
Sh4/51
__
Hence
max{ le'(xi1)l e'(xi+1)() h4/25)
contradicting the fact (4.3.17) that
h4/61 1 max ( e'(xi1), e'(xi+1) 3
We have shown by assuming the contrary that
e'(x ) 0 for i = 1, 2, ..,k 1.
Condition (4.1.3) is that e' (x0) and e' (xk) are zero. Thus
(4.3.33) e'(x ) ( 0 for i = 0, 1 ., k.
Applying (4.3.32) again we have for i = 1, 2, ., k 1
Se'(xi) = h4/5! + e'(xi1) + e'(xi+1)
As e'(xi1), e'(xi+1) i 0, this implies that
8e'(x ) ( h4/5!
and
(4.334)e'(x ) < h4/[8(5!)]
Similarly, for i = 2, 3, ., k 2, we have
8e'(xi) = h4/5! + e'(xi1) + e'(xi+1) '
and hence by (4.3.34),
e'(x ) 4/5 h4/[8(5!1) h4/[8(5!1)
=(1 + 1/4) h4 /[8(51)].
Inductively, for i = j to i = k j, we will have
e'(x ) L {1 + 1/4 + 1/42 + .. + 1/4j13h4/[8(5 1)]
The harmonic series (1 + 1/4 + 1/2 + ..)i
equal to 1/(1 1/4) or 4/3 Thus, in the limit as i, k,
and j go to infinity, we have
(4.3.35) e'(x ) I (1/8)(4/3)h4/5! = h4/6!.
We already know from (4.3.17) that Je'(x )] J4/6.Tu
for k > 2j + 1, and k j > i > j as j goes to infinity,
we have
(4.3.36) e'(x ) goes to h4/6!.
In the sense of (4.3.36), (4.3.31) is satisfied.
Then, as e'(xi ) goes to h4/! x55 sx)ge
uniformly to the expression of (4.3.30) and (4.3.20) with
h = hi. It follows that the expression of (4.3.20)
cannot be improved further. In fact we have shown that
c(t) offers a pointwise exact bound, and its maximum co
is the exact norm bound.
Proof of Theorem 4.2
We know from (4.3.1) that for xi < x < xi+1'
(4.4.1) s(x) f(x) = Pl(t) f(xi) + 2(t) f(zi+1)
+ P3(t) f(xi+1) + hi 4q(t) s'(xi)
+ hi 5S(t) s'(xi+1) f(x).
It is easily verified that P (t) + P2(t 3 (t) = 1.
Thus
(4.4.2) s(x) f(x) = P1(t) If(x ) f(x)}
+ P2(t) [f(zi+1) f(x)]
+ P3(t) [f(xi+1) f(x)]
+ hi s'(x ) P4 t)
+ hi s'(xi+1) 5S(t).
Each of the first three terms on the right hand side
can be bounded in absolute value by (f,h). We also must
bound the last two terms. For equally spaced knots h = hi
= h1;equation (4.1.4) reduces to
(4.4.3) h s'(xi1) + 8h s'(x ) h s'(xi+1)
=16 [f(zi+1 f(zi)l
5 [f(xi+1) f(xi1 *
Assume s'(xi) is maximal in absolute value. Then
6h s'(xi)l ( 16 f(zi+1) f(zil
+ 5 If(xi+1) f(xi)
+ 5 If(xi) f(xi1)
( 26 w(f,h),
and hence
(4.4.4) Ih s'(x ) ( (13/3) w(f,h).
Combining (4.4.2) and (4.4.4), we have
(4.4.5) Is(x)f(x)( I IIP (t)J + IP (t)l + IP (t)
+ (13/3)IP4(t)l + (13/3)IP4(t)( } 0(f,h).
For 0 < t < 1/2,
(4.4.6) Pltl I2() I3(t
+ (13/3) P4(t)j + (13/3) P4~~
= Pl(t +2(t P3( +
(13/3)P4(t) + (13/3)P5(t)
=1 + (13/3)t 3t2 (58/3)t3 + 16 t4
We have shown the theorem for 0 < t <1/2. The
argument for 1/2 < t < is symmetric.
Proof of Theorem 4.3 We are considering now the case in
which knots are no longer assumed to be equal. We assume
that the ratio of the longest subinterval to the shortest
is less than m. Equation (4.4.2) still applies. Again we
choose i so that Is'(xi)l is maximal. From (4.1.4) we now
have
(4.4.7) his'(xi1) + 4(hi+hi1) s'(xi) hi1s'(xi+1)
= 1 (i1h i ih) h
+ 16 [(hi1/hi) f(zi+1) h i/hi1) f(zi)l
5 [(hi1/hi) f~xi+) (hi/hi1) f(xi1)]
= 11[hi/hi11 [f(xi) f(zi)l
+ 5[hi/hi11 1f(zi) + f~xi1~
+ 11[hi1/h ] If(zi+1) f(xi)l
+ 5[hi1/hi] [f(zi+1) f(xi+1 *
Then
3 (hi + hi1) js'(xi)J 16 hi/hi1 w(f, hi1/2)
+ 6hi1/hi o(f,hi/2)
and
Is'(xi) ( (16/3) [hi/hi1 + hi1/hi] w(f,h)
hi + hi1
where h = max0
Then for any given j, O < j < k1 and
m = maxlhi)/{mini), i = 0, 1, . ., k 1, we have
(4.4.8) max{ hj s'(x ) (h s'(xj+1)l
< (16/3) h (m + 1/m) w(f,h)
2min~hi~
< (8/3) (m2 + m) w(f,h).
Substituting (4.4.8) into (4.4.2) yields the result
of the theorem.
CHAPTER FIVE
IMPROVED ERROR BOUNDS FOR THE PARABOLIC SPLINE
Introduction and Statement of Theorems
The quartic splines of Chapter Four share and improve
many of the properties of the complete cubic spline. To
insure a good approximation to a given continuous
function, we must make the largest subinterval of a
partition small. Unfortunately, we must also pose some
additional restrictions on the partition. For instance,
in Theorem 4.3, the norm of the error depends not only on
the length h of the largest subinterval but also on the
ratio m of the largest to smallest length subinterval.
Similar additional restrictions must be made for the cubic
spline.
In this chapter, we will discuss a spline operator
for which the norm of the approximation error goes to zero
with the length of the largest subinterval, for any par
tition and any continuous periodic function. This spline
is the piecewise parabolic spline introduced by Marsden
and discussed in Chapter One. Its properties are
summarized in Equations (1.6.1) to (1.6.7).
As Marsden points out, many of the bounds he gives
can be sharpened. The main result of this chapter will be
to accomplish this sharpening. While many of the bounds
given here may still not be exact, at least one of them
is, and in fact is even pointwise exact. In other cases we
can reduce the known bounds by a factor of more than two.
The results given here thus enable one to compare the
error of the Marsden spline to the error of other spline
interpolation processes. Specifically, future work on the
cubic spline interpolant should shed light on the validity
of Marsden's conjecture that the parabolic spline offers
better approximation than the cubic spline when functions
of the classes C(1), C(2), and C(3) are considered.
We first recapitulate the properties of the parabolic
spline. Let
f: 8 Cla,b] ,f(a) = f(b),
Ilf = sup ([f(x)l : a < x < b }
such that f is extended periodically with period b a.
A function s(x) is defined to be a periodic quadratic
spline interpolant associated with f and a partition
{xi ~=0
a) s(x) is a quadratic expression on each (xi1, xi)
b) s(x) 6 C'[a,10];
c) s(a) = s(b) s'(a) = s'(b)
d) s(zi) = f(zi) i = 1, 2 ., k
where zi+1 = (xi+1+x )/2.
The following theorem is due to Marsden [1974] and
was given in Chapter One as Theorem 1.13.
Theorem Let {x ) =0 be a partition of la, b], f(x) be a
continuous function of period b a, and s(x) be the
periodic quadratic spline interpolant associated with f
and {xi=,0. he
(5.1.2) jjsil ( 2 I((fj i)"s L 2 Jf I
jje.l I 2 w(frh/2);
jje II 3 w(f,h/2).
(where si = s(xi) and ei = y(xi) s(xi) *
The constant 2 which appears in the first of the above
equations can not, in general, be decreased.
For continuous functions to be "wellapproximated" by
the spline s, Equations (5.1.2) show that the only
requirement for the partition is that the length h of the
largest subinterval be small enough that the modulus of
continuity of f be small.
Concerning s, we can prove the following results.
These are analogous to the results of Marsden given above
as Theorems 1.14 to 1.16 and improve upon the bounds he
derived.
Theorem 5.1 Let f and f' be continuous functions of period
b a. Then
(5.1.3) je(x) < c~ h Ilf' ,
whee a =2/3 J1fl/6 and 001 1 +a0 8a02 + 4a03 or
C0,1 is approximately 1.0323. The analogous constant from
Marsden was 5/4.
Theorem 5.2 Let f, f', and f" be continuous functions of
period b a. Then
(5.1.4) e ( (1/6) h2 Il.~
(5.1.5) I(e 'l( < (9/16) h f ,
(5.16) e'l ( (17/16) h f" .
(Marsden's constant for (5.1.4) was 5/8, while in (5.1.6)
the value was 2).
If we make the additional assumption that the
partition consists of equally spaced intervals, then we
can improve (5.1.6) to
(5.1.7) e'l ( .7431 h f" 
Theorem 5.3 Let f, f', f", and f"' be continuous
functions of period b a. Then
(5.1.9) (eil < (1/24) h3 Il"'l
(51.0)e < (1/6) h2 llf'l
(5.1.11) (e( (1/24) h3 ..f'(
(51.2)e'l (7/24) h2 Ils'
jje"l [h /2 + (h3/3hi2)] If"'" xi1
Marsden's analogous constants for (5.1.9) to (..1
are 1/8, 1/3, 17/96, and 11/24 respectively.
Furthermore, (5.1.9) and (5.1.11) are best possible.
In fact we also have the exact pointwise bound
(5.1.14) (e(x)l < IE3(t)( h3 lif" 'lJ, xi < x < xi~ ,
where t = (x x )/(xi+1 x ) and
Q3(t) = 1/24 t2/4 + t3/6
is the "Euler spline" of degree 3.
The technique used here is the same as that used in
the last chapter. For a given partition subinterval
[xj, xi+1], we write
(5.115) f )x) s () If i (x) L )(x)
+ L (x) s i)(x)
where L is a polynomial interpolation of f. We then
proceed by obtaining pointwise estimates of the quantities
on the right hand side of (5.1.15).
Proof of Theorem 5.1
Given that f and f' are continuously differentiable
of period b a, we will establish the following pointwise
bound for the parabolic continuously differentiable spline
s interpolating function values at subinterval midpoints
zi. Let xi < x < xi+1 and let
t = (x xi)/(xi+1 x ) and h = max~hi *
Then for xi < x < zi+ we have
(5.2.1) jf(x) s(x)l I h [1 + t 8t2 + 4t3] Il'
For z ~ < x <_ xi+1 replace t in (5.2.1) by 1 t.
Equation (5.1.3) follows from (5.2.1).
In order to establish (5.2.1) we write for
xi < x < xi+1'
f(x) s(x) = f(x) L(x) + L(x) s(x)
where L(x) is the parabola matching f(xi), f(zi+1), and
f(xi+1). Then
(5.2.2) (f(x) s(x) I (f(x) L(x) + JL(x) s(x))
We can represent L(x) as
(5.2.3) L(x) = f(x )AO(t) + f(zi+1)A1(t) + f(xi+1)A2 t)
where t = (x xi)/hi and
AO (t) = 2 (12t(1t
Al(t) = 4t (1 t)
A2(t) = 2t (t 1/2).
As L reproduces parabolas exactly and as the restriction
of s(x) to [xi, xi+11 is a parabola, for xi < x < xi+1 we
have
(5.2.4) s(x) = s(x )AO(t) + s(zi+1)A1(t) + s(xi+1)A2 *)
As f(zi+1) = s(zi+1), we have
(5.2.5) L(x) s(x) ( jf(x ) s(xi ) AO(t)
+ f(xi+1) s(xi+1)( A2(t
L { AO~t) + IA2(t) 3 leil
< )1 2tJ jje
where lei = max1
We have shown that
(5.2.6) f(x) s(x)
f(x) L(x) + 1I 2tl ei *
It remains to bound If(x) L(x) and lei in terms of
Marsden showed that
(5.2.7) je  < h jf'j
where h is the maximum length of a subinterval.
In order to bound f(x) L(x) we resort to the
Peano theorem. Defining g(t) =: f(xi+hit) = f~x), we have
(5.2.8) f(x) L(x) = I_ K1(t,z)g'(Z)dz
where
K1(tlz) = (tz)0 AO(t) 10z]
A1(t) [1/2 z]0,
A2 (t) (1 z]0
and
1 for t > z
(tz) =
0 for t < z.
In order to verify (5.2.8), one need only expand the right
hand side and integrate by parts. For 0 < t < 1/2,
K (t,z) may be written in the more convenient form
K (t,z) = A (t) A (t) + 1 for 0 < z < t
=A (t) A2 (t) for t < z < 1/2
= A (t) for 1/2 < z < 1.
From Equation (5.2.8) it follows that
(5.2.9) If(x) L(x) ( I_ Kl(t,z) dz max0
L hi I IKl(t,z) Idz maxx.
i 1~
0h/ ~~~~d ~'
Evaluating the integral in (5.2.9), we have
(5.2.10) 1_ Kl(t,z) dz = t (1 Al(t) A2~~
+ (1/2 t) [A1(t) + A2(l
+ 1 1/)[A2 ()
=3t 8t2 + 4t3
Combining Equations (5.2.7)(5.2.10) we have for
0 < t <1/2,
(5..11 f (x) s (x)l L (h (1 2t) +
hi (3t 8t2 + 4t 3
( h [1 + t 8t2 + 4t3
which is precisely the desired result. The maximum of the
right hand side of (5.2.11) occurs for a0 = 2/3 13/6.
Evaluating gives the value C0,1"
Proof of Theorem 5.2
Let f be twice continuously differentiable of period
b a and let a partition
a = x0 < "1 < x1 < . xi < zi+1 < xi+1 <. < xn = b
be given (where zi+1 = (xi + xi+1)/2, every i). Let s be
continuously differentiable and a parabola on each
interval [x xi+1] such that
s(zi+1) = f(zi+1), s(a) = s(b), and s'(a) = s'(b).
Letting t = (x x )/h we show that
(5.3.1) jf(x) s(x)( < c l(t) Ilf"
CO,2(t) = h2 ((1 2t)/6 + [ t/(3 2t) t2]
and for zi+1 < x < xi+1
c0,2(t) = c0,2(1t).
Furthermore the maximum of c0,2(t) is 1/6 and occurs
for t = 0 and 1.
As in the proof of Theorem 5.1 we fix i and let L(x)
be the parabola satisfying
L(x )=f(x ) L(zi+1)= f(zi+1)
L(xi+1) = f(xi+0 *
Then, proceeding in the same way as before,
(5.3.2) f(x) s(x)) < If(x) L(x)( + jjeil (1 2t .
We must bound If(x) L(x) and Ilei . We first
bound j ei1. From Marsden [1974), we have
(5.3.3) hi si1 + 3(hi + hi1) si + hi1 si+1
=4 [hi f(z ) + hi1 f(zi+1) *
Denoting fi =f(xi) and ei = si ,we obtain from
Equation (5.3.3)
(5.3.4) hi ei1 + 3(hi + hi1) ei + hi1 ei+1
=hi fi1 4 hi f(zi) + 3(hi + hi1 fi
4 hi1 f(zi+1) + hi1 fi+1
=:B(f) .
As B is identically zero for any linear function f,
we have by the Peano Theorem:
(5.3.5) B(f) = xi+1K(y) f" (y) dy / 1!
xi1
where
K(y) = B [(xy) ]
= hi1 (xi+1 Y+ 4hi1 (zi+1 Y+
+ 3(hi + hi1) (xi Y+ 4hi (Zi Y+
+ hi (xi1 Y+
and x y for x y
(x y) =
0 for x < y.
In order to illustrate the symmetry of the kernel
K(y) about xi, we expand in terms of y xi to obtain
K(y) = hi1 [hi (yxi)l
for h /2 I yxi < hi
=hi1 [3(yxi) hi]
for 0 L yxi < hi/2
=hi [3(yx ) hi1]
for h _/2 I yxi < 0
=hi [hi1 + (yxi)l
for hi1 I y xi I hi1/2
whee i = xi+1 xi and hi1 = xi xi1. As is easily
seen, the sign of K(y) changes at y = xi + hi/3 and
xi hi1/3.
From (5.3.5), it follows that
(5.3.6) Ihi ei1 + 3(hi + hi1) ei + hi1 ei+1l
xi+1
< I IK(y) dy f"
xi1
( (hi + hi1) hi hi1 llf" j/3
Let i be such that leil = Iei . Then
(5.3.7) j e.l ( (1/6) h2 If
which is the desired bound on I ei~l
We next bound If(x) L(X) where L is the parabola
matching f at x zi+1, and xi+1. L can be uniquely
expressed as
(5.3.8) L(x) = f(x ) AO(t) + f(zi+1) Al~t
+ f(xi+1) A2(t)
where
AO (t) = 2 (/ )1t
A (t) = 4t (1 t),
A2(t) = 2t (t 1/2).
Then, defining g(t) =: f(xi+hit) = f(x) we have
(5.3.9) f(x) L(x) = J K2(t,z) g" (z)dz t = (xx )/hi
where
K2(t,z) = (t 2), AO(t)[0z],
1 A(t) [1/2 z], A2(t) [1 z] .
Equation (5.3.9) can be verified by integrating by parts
to obtain (5.2.8). For 0 < t < 1/2, K2 takes the form
(5.3.10) K2(t,z) = 2 (2t 1)(1 t) for t >z, t < 1/2,
t [1 + z(2t 3)] for t < z ( 1/2
t (2t 1) (1 z) for 2 >1/2, t < 1/2.
From (5.3.9), it follows that for 0 < t < 1/2
(5.311)f(x) L(x)l/l g" (t)l L r IK2(tlz) dz
= z(2t 1)(1 t) dz
1/(32t)
t[1 + z(2t 3)] dz
1/2
+ J t~l + z(2t 3)] dz
1/(32t)
2t(t 1/2)(1 z)dz
1/2
= t2 + [t/(32t)]
Therefore if 0 < t < 1/2, we have
(5.3.12) If(x) L(x)J < h 2 Et2 + [t/(32t)]}Jf" (x))
We can now assemble the parts to get' the pointwise bound
(5.31).Using the bound for ]f(x) L(x) of (..1
and the bound of (5.3.7) for ei  in the formula
Jf(x) s(x)l L If(x) L(x)( +  ejj 1 2t ,
we then have for 0 < t < 1/2,
(5.3.13) If(x) s(x)l ( { hi2 [t2 + t/(32t)]
+ (1 2t) h2/6 ) I f"
which, as hi I h, immediately implies (531.The result
for 1/2 < t < 1 follows by symmetry. It remains only to
be shown that the maximum of
0 < c ,(t) = h2 ((1 2t)/6 + [ t/(3 2t) t2~
is h2/6 and occurs for t = T e hs xad0,
at 0 as
c0,2(t) = C002(0) + t '0,2'(0) + (t2/2) c0,2 7)Y
where 0 < t < 1/2 and 0 < y < t. It is not hard to
verify that the last two terms of the above expression are
negative, and hence the maximum occurs at t = 0. This
completes the proof of Equation (..)
We next show Equation 5.1.5. From Marsden, we have
the tridiagonal system matching spline derivatives,
(5.3.14) hi1 si1' + 3(hi + hi1) si' + hi si+1l
=8 [f(zi+1) f(zi)l
or equivalently,

