Table Of ContentSpringer Tracts in Natural Philosophy
Volume 13
Edited by B. D. Coleman
Co-Editors: R.Aris· L.Collatz· J.L.Ericksen· P.Germain
M. E. Gurtin . M. M. Schiffer' E. Sternberg· C. Truesdell
Gunter Meinardus
Approximation of Functions:
Theory and Numerical Methods
Translated by Larry L. Schumaker
Springer-Verlag New York Inc. 1967
Expanded translation of the German version:
Approximation von Funktionen und ihre Numerische Behandlung.
Springer Tracts in Natural Philosophy, Volume 4
Professor Dr. Gunter Meinardus
Institut flir Mathematik
der Technischen Hochschule Clausthal
ISBN 978-3-642-85645-7 ISBN 978-3-642-85643-3 (eBook)
DOI 10.1 007/978-3-642-85643-3
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procedure without written permission from the Publishers. © by Springer-Verlag Berlin. Heidelberg 1967.
Softcover reprint of the hardcover 1st edition 1967
Library of Congress Catalog Card Number. 67-21464.
Title No. 6741
Preface to the first German edition
It has only been in the past few years that those parts of approxima
tion theory which can be applied to numerical problems have been
strongly developed. The idea of obtaining a (in some sense) best approxi
mation of a function gained considerable importance with the application
of electronic computers. Some of the theoretical fundamentals necessary
for practical problems can be found scattered about in a few books.
However, by far the greatest portion of the theoretical and practical
investigations can be studied only in the original papers. This provides
the purpose of this book: to collect essential results of approximation
theory which on the one hand makes possible a fast introduction to the
modern development of this area, and on the other hand provides a
certain completeness to the problem area of Tchebycheff approximation
not to imply by any means that a comprehensive survey of the literature
is attempted. The material has been chosen from the subjective stand
point of its importance for applications. This also applies, for example,
to the asymptotic investigations of § 3, since I am of the opinion that
even in numerical approximation some thought should at least be given
to what asymptotic precision can be expected. I have confined myself
almost exclusively to the theory of uniform approximation since it has
by far the greatest practical importance.
Part I is concerned with linear approximation. Chapter 3 contains
what at present must be considered as the shortest approach to the
linear theory. The details of the classical case of polynomial approxima
tion (§ 6) are not much known, and the approach to the results is often
laborious, so that I have decided to give a complete exposition. A special
chapter (§ 7) has been dedicated to numerical methods of linear approxi
mation, while constructive methods for non-linear approximation have
been included with the theory in the individual sections. The bulk of
Part II is related to newer investigations which I have carried out with
D. SCHWEDT. Here we develop a theory of non-linear approximation
which can be applied to various numerical problems.
With a few exceptions, all of the theorems in normal type have been
presented with proofs (partly new). References to further studies are
set in small type.
Unfortunately, because of space limitations various aspects of
approximation theory have been completely disregarded. This includes,
VI Preface to the English edition
for example, the so-called Lp approximation, the Bernstein approxima
tion problem (approximation on the real line by certain entire functions),
and the highly interesting studies of J. L. WALSH on approximation in
the complex plane.
I would like to extend sincere thanks to Professor L. COLLATZ for
his many encouragements for the writing of this book. Thanks are
equally due to Springer-Verlag for their ready agreement to my wishes,
and for the excellent and competent composition of the book. In addition,
I would like to thank Dr. W. KRABS, Dr. A.-G. MEYER and D. SCHWEDT
for their very careful reading of the manuscript.
Hamburg, March 1964
GUNTER MEINARDUS
Preface to the English Edition
This English edition was translated by Dr. LARRY SCHUMAKER,
Mathematics Research Center, United States Army, The University of
Wisconsin, Madison, from a supplemented version of the German edition.
Apart from a number of minor additions and corrections and a few new
proofs (e.g., the new proof of JACKSON'S Theorem), it differs in detail
from the first edition by the inclusion of a discussion of new work on
comparison theorems in the case of so-called regular Haar systems (§ 6)
and on Segment Approximation (§ 11). I want to thank the many readers
who provided comments and helpful suggestions. My special thanks are
due to the translator, to Springer-Verlag for their ready compliance
with all my wishes, to Mr. HELMUT UNTERSTEIN for his valuable help,
and to Miss GUDRUN STECHER and Miss CHRISTEL FRANKE for their
careful typing of the manuscript.
Clausthal-Zellerfeld, May 1967
GUNTER MEINARDUS
Contents
Part 1. Linear Approximation
§ 1. The General Linear Approximation Problem.
1.1. Statement of the Problem. Existence Theorem 1
1.2. Strictly Convex Spaces. Hilbert Space . 2
1.3. Maximal Linear Functionals 4
§ 2. Dense Systems. . . . . . . . . 5
2.1. A General Criterion of BANACH 5
2.2. Approximation Theorems of WEIERSTRASS and MUNTZ. 6
2.3. Approximation Theorems in the Complex Plane. 10
§ 3. General Theory of Linear Tchebycheff Approximation. 13
3.1. Fundamentals. The Theorem of KOLMOGOROFF . 13
3.2. The Haar Uniqueness Theorem. Linear Functionals and Alter-
nants . . . . . . . . . . 16
3.3. Further Uniqueness Results. 24
3.4. Invariants . . . . . . . . 26
3.5. Vector-valued Functions. . 28
§ 4. Special Tchebycheff Approximations 28
4.1. Tchebycheff Systems . . 28
4.2. Tchebychef£ Polynomials. . . 31
4.3. The Function (x - a)-l. . . . 33
4.4. A Problem of BERNSTEIN and ACHIESER 36
4.5. ZOLOTAREFF'S Problem . . . . . . . 41
§ 5. Estimating the Magnitude of Error in Trigonometric and Polynomial
Approximation. . . . . . . . . . . . . . . . . . . . . . . . 45
5.1. Projection Operators. Linear Polynomial Operators . . . . . 45
5.2. The Connection between Trigonometric and Pol ynomial A pprox-
imation . . . . . . . . 45
5.3. The Fejer Operator . . . . . 47
5.4. The Korovkin Operators. . . 50
5.5. The Theorems of D. JACKSON. 52
5.6. The Theorems of BERNSTEIN and ZVGMUND. 57
5.7. Supplements . . . . . . . . . . . . . . 65
§ 6. Approximation by Polynomials and Related Functions 72
6.1. Foundations . . . . . 72
6.2. Upper Bounds for En (f) . . . . . . . . . . . 77
6.3. Lower Bounds for En (f) . . . . . • . . . . • 82
6.4. Dependence of the Approximation on the Interval. 85
6.5. Regular Haar Systems. . 87
6.6. Asymptotic Results . . . 90
6.7. Results for the Alternants 101
VIII Contents
§ 7. Numerical Methods for Linear Tchebychef£ Approximation 105
7.1. The Iterative Methods of REMEZ 105
7.2. Initial Approximations. . . . 116
7.3. Direct Methods ...... . 122
7.4. Discretization. Other Methods. 124
Part II. Non-linear Approximation
§ 8. General Theory of Non-linear Tchebychef£ Approximation 131
8.1. Survey of the Problem. A Generalization of the Kolmogoroff
Theorem. . . . . . . . . . . . . . . . 131
8.2. The Haar Uniqueness Theorem. Alternants. 141
8.3. The Investigations of RICE. . 148
8.4. The Newton Iteration Method 149
8.5. H-Sets. . . . . . 153
§ 9. Rational Approximation 154
9.1. Existence. Invariants. A Theorem of WALSH 154
9.2. Theorems on Alternants. Anomalies. Continuity. Examples 160
9.3. Asymptotic Results. Small Intervals. 167
9.4. Numerical Methods . 170
§ 10. Exponential Approximation 176
10.1. The Results of RICE. 176
10.2. An Anomaly Theorem. Constructive Methods. 179
§ 11. Segment Approximation. . . . . . . . . . 183
11.1. Statement of the Problem. Hypotheses 183
11.2. The principle of LAWSON ••••••• 184
11.3. Equi-degree Polynomial Approximation 188
Bibliography 189
Subject Index. 197
Part I
Linear Approximation
§ 1. The General Linear Approximation Problem
1.1. Statement of the Problem. Existence Theorem. Let R be a normed
linear space of elements I, g, ... over the field of real or complex numbers,
and let the symbol IIIII denote the norm of I. In addition, let V be a
finite dimensional linear subspace of R. The general linear approxima
tion problem is as follows:
For a given IE R determine an element gE V such that
Ilg- Ilh-III
III~ (1 .1)
for all hE V.
Theorem 1: For any given fE R there exists a gE V satisfying property
(1 .1).
Proof (R. C. BUCK [1J; N. I. ACHIESER [1J, p. 10): Setting
IIh-/ll,
ev(l) = inf
hEV
the inequality
Ilh-fll> Ilfll
~ ev(l)
Ilhll>211/11.
holds for all h satisfying Hence it suffices to consider the
IIh-/11
continuous function on the set of all hEY with Ilhll~211/11. Since
this set is a closed and bounded subset of the finite dimensional space V,
Ilh-III
it is compact, and consequently assumes its minimum.
An element gE V satisfying
Ilg-111=
(Iv (I)
is called a best approximation of I with respect to V. Obviously the set of
best approximations is convex. The above existence theorem has been
proved under more general assumptions by G. KOTHE [1J, p. 347.
The functional ev (I) is a seminorm, and since
I I 11/ f211,
(Iv (11) - (Iv (12) ~ (Iv (/1-12) ;;::; 1-
it is also a continuous function of f.
1 Springer Tracts. Vol. 13. Meinardus
2 § 1. The General Linear Approximation Problem
1.2. Strictly Convex Spaces. Hilbert Space. Following M. KREIN [1],
we call the space R strictly convex (or strictly normed) if
11/+gll<2 whenever 11/11=llgll=1 and I=t=g. (1.2)
Then the following uniqueness theorem holds:
Theorem 2: II R is strictly convex, then lor each IE R and each linite
dimensional subspace V (R there exists precisely one best approximation
01 I with respect to V.
Cf. R. C. BUCK [lJ; N. 1. ACHIESER [lJ, p. 11; G. KOTHE [lJ, p. 347.
Proof: If gl and g2 are both best approximations, gl =t= g2' then it
follows from (1.2) that
Iii (gl + g2) - III = ill (gl-I) + (g2-I) II < ev (I)
in contradiction with the definition of ev(l).
Fig. 1. The unit ball corresponding to a strictly convex and a non-strictly convex
norm in two dimensions
The converse of Theorem 2 also holds: If R is not strictly convex,
then there is some V (R and an IE R such that there exists more than
one best approximation of I with respect to V. For generalizations of
this result see G. KOTHE [1J.
Hilbert space and the spaces lp and Lp for 1 < p < are examples
00
of strictly convex spaces (G. KOTHE [lJ). Existence theorems in other
spaces, e.g., in the space L1[a, bJ of integrable functions I(x) on the
interval [a, bJ with norm
b
11/11=fl/(x)ldx, (1.3)
a
or in the space C[a, bJ of continuous functions I(x) on the interval
[a, bJ with norm
11I11 = max It(x) I, (1.4)
xE[a, b]
depend on the special subspace V under consideration (d. § 3.2), and
frequently even on special properties of the function f itself (d. § 3.3).
1.2. Strictly Convex Spaces. Hilbert Space 3
Introducing a basis hi' h2' ... , h for the subspace V, we see that
n
is a real-valued function of the (real or complex) numbers IX •• This
function is continuous in the IX., but in general is not differentiable
everywhere.
In a Hilbert space with the inner product (I, h), the strict convexity
follows directly from the parallelogram law
111+ g1 12+ III - g1 12= 211/112 + 211g 11
2•
The best approximation g of I with respect to V is then the orthogonal
projection of I on V, i.e.,
(g-I, h)=O
for all hE V. This can be shown as follows: If there exists an hoE V such
that
(g-I, ho)=f3~O,
then clearly
- {J
gl = g- (ho' hol . ho
lies in V, and
But this contradicts the fact that g is a best approximation.
The best approximation g can be obtained directly from the above
projection property. Indeed, if the elements hi' h2' ... , hn form an
orthonormal basis for V, then
n
g= L
(I, h.) h•
• =1
and moreover,
n
11/-g[l2= 11/112-L I 1
(I, h.) 2 •
• =1
If the elements g1' g2' ... , gn form a basis for V which is not necessarily
orthonormal, then
where the A. can be obtained from the linear system of equations
n
L J...(g., gl')= (I, gl')' fl=1, 2, ... , n.
,=1
