n-Widths in Approximation Theory

300 Pages·1985·8.78 MB·English

by  Allan Pinkus (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete
3. Folge . Band 7
A Series of Modern Surveys in Mathematics
Editorial Board
E. Bombieri, Princeton S. Feferman, Stanford
N. H. Kuiper, Bures-sur-Yvette P. Lax, New York
R. Remmert (Managing Editor), Munster
W Schmid, Cambridge, Mass. J-P. Serre, Paris
J. Tits, Paris
Allan Pinkus
n-Widths in
Approxilllation Theory
Springer-Verlag
Berlin Heidelberg New York Tokyo 1985
Allan Pinkus
Technion
Israel Institute of Technology
Department of Mathematics
Haifa 32000, Israel
AMS Subject Classification (1980): 41-02, 41A46, 41A65
ISBN-13 :978-3-642-69896-5 e-ISBN-13: 978-3-642-69894-1
DOl: 10.1007/978-3-642-69894-1
Library of Congress Cataloging in Publication Data
Pinkus, Allan, 1946- N-widths in approximation theory.
(Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 7)
Bibliography: p. Includes index.
1. Approximation theory. I. Title. II Series.
QA221.P56 1985 511'.4 84-13902
ISBN-13 :978-3-642-69896-5 (U.S.)
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To Rachel
Preface
My original introduction to this subject was through conservations, and ultimate
ly joint work with C. A. Micchelli. I am grateful to him and to Profs. C. de Boor,
E. W. Cheney, S. D. Fisher and A. A. Melkman who read various portions of the
manuscript and whose suggestions were most helpful. Errors in accuracy and
omissions are totally my responsibility. I would like to express my appreciation
to the SERC of Great Britain and to the Department of Mathematics of the
University of Lancaster for the year spent there during which large portions of the
manuscript were written, and also to the European Research Office of the U.S.
Army for its financial support of my research endeavors. Thanks are also due to
Marion Marks who typed portions of the manuscript.
Haifa, 1984 Allan Pinkus
Table of Contents
Chapter I. Introduction . . . . . . . . 1
Chapter II. Basic Properties of n-Widths . 9
1. Properties of dn • • • • • • • • • • 9
2. Existence of Optimal Subspaces for dn • 15
3. Properties of dn • • • • • • 17
4. Properties of b 20
n • • • • • •
5. Inequalities Between n-Widths 22
6. Duality Between dn and dn • • 27
7. n-Widths of Mappings of the Unit Ball 29
8. Some Relationships Between dn(T), dn(T) and bn(T) . 32
Notes and References . . . . . . . . . . . . . . 37
Chapter III. Tchebycheff Systems and Total Positivity 39
1. Tchebycheff Systems 39
2. Matrices . . . . 45
3. Kernels. . . . . 52
4. More on Kernels . 56
Chapter IV. n-Widths in Hilbert Spaces 63
1. Introduction. . . . . . . . . . . 63
2. n-Widths of Compact Linear Operators 63
3. n-Widths, with Constraints . . . . . 73
3.1 Restricted Approximating Subspaces 73
3.2 Restricting the Unit Ball and Optimal Recovery. 77
3.3 n-Widths Under a Pair of Constraints . . . . . 80
3.4 A Theorem of Ismagilov . . . . . . . . . . . 93
4. n-Widths of Compact Periodic Convolution Operators 95
4.1 n-Widths as Fourier Coefficients . . 95
4.2 A Return to Ismagilov's Theorem . . . 98
4.3 Bounded mth Modulus of Continuity . 101
5. n-Widths of Totally Positive Operators in L2 107
5.1 The Main Theorem . . . . . . . . 108
5.2 Restricted Approximating Subspaces 117
X Table of Contents
6. Certain Classes of Periodic Functions . . . . . . . . . 125
6.1 n-Widths of Cyclic Variation Diminishing Operators. 126
6.2 n-Widths for Kernels Satisfying Property B . 128
Notes and References . . . . . . . . . . . . 136
Chapter V. Exact n-Widths of Integral Operators 138
1. Introduction. . . . . . . . . . . . . 138
2. Exact n-Widths of .Yt'oo in Lq and ~ in L1 139
3. Exact n-Widths of .Yt':, in Hand .Yt'; in V 154
4. Exact n-Widths for Periodic Functions 169
5. n-Widths of Rank n + 1 Kernels 188
Notes and References . . . . . . 195
Chapter VI. Matrices and n-Widths 198
1. Introduction and General Remarks 198
2. n-Widths of Diagonal Matrices. . 202
2.1 The Exact Solution for q ~ p and p = 1, q = 2 202
2.2 Various Estimates for p = 1, q = 00 . • . 212
3. n-Widths of Strictly Totally Positive Matrices 223
Notes and References . . . . . . . . . . . . 231
Chapter VII. Asymptotic Estimates for n-Widths of Sobolev Spaces 232
1. Introduction. . . . . 232
2. Optimal Lower Bounds . . . 234
3. Optimal Upper Bounds. . . 236
4. Another Look at <>n(Br); Loo) . 241
Notes and References . . . . . 246
Chapter VIII. n-Widths of Analytic Functions 248
1. Introduction. . . . . . . . . . . . . . 248
2. n-Widths of Analytic Functions with Bounded mth Derivative 249
3. n-Widths of Analytic Functions in H2 . 258
4. n-Widths of Analytic Functions in HOO 263
5. n-Widths of a Class of Entire Functions 270
Notes and References 275
Bibliography. . . . 277
Glossary of Selected Symbols. 288
Author Index 289
Subject Index. 291
Chapter 1. Introduction
In this short chapter we introduce the subject matter. We hope that this introduc
tion whets the reader's appetite for the more systematic treatment which will
follow.
Let X be a normed linear space and Xn any n-dimensional subspace of X. For
each x E X, E (x; Xn) shall denote the distance of the n-dimensional subspace Xn
from x, defined by
If there exists a y* E Xn for which E (x; X n) = II x - y* II , then y* is a best approxi
mation to x from Xn. Problems of existence, uniqueness, and characterization of
the best approximation are of central importance in approximation theory. (Since
Xn is here a finite dimensional subspace of X, a best approximation always exist.)
For example, consider X = C [a, b], the space of real-valued continuous functions
on the finite interval [a, b], endowed with the uniform norm. The classical Haar
Theorem delineates those n-dimensional subspaces of C [a, b], called Chebyshev
systems, for which there is a unique best approximation to every fEe [a, b]. This
unique best approximant is characterized by the fact that the error function
equioscillates on at least n + 1 points in [a, b].
Let us now suppose that instead of a single element x, we are given a subset
A of X. How well does the n-dimensional subspace Xn of X approximate the
subset A? A commonly used definition is to set
E(A;Xn) = sup {E(x;Xn): x E A}
= sup inf II x - y II.
xeA yeXn
E(A;Xn) is the deviation of A from Xn. Thus E(A;Xn) measures the extent to
which the "worst element" of A can be approximated from Xn. Many results in
approximation theory are concerned with this particular quantity for specific
choices of A and X
n •
Given a subset A of X, one might also ask how well one can approximate A
by n-dimensional subspaces of X. Thus, we shall consider the possibility of allow
ing the n-dimensional subspaces Xn to vary within X. This idea was first pro
pounded by Kolmogorov [1936].
Definition 1. Let X be a normed linear space and A a subset of X. The n-width,
in the sense of Kolmogorov, of A in X (or the Kolmogorov n-width of A in X) is
2 I. Introduction
given by
dn(A;X) = inf{E(A;Xn): Xn an n-dimensional subspace of X}
= inf sup inf II x - y Ilx,
x" xeA yeXn
the left-most infimum being taken over all n-dimensional subspaces Xn of X.
(Some authors prefer the expression "n-diameter" rather than "n-width". We
shall always use the latter term.)
A subspace Xn of X of dimension at most n for which
is called an optimal subspace for d (A; X).
n
Since d (A; X) measures the extent to which A may be approximated by
n
n-dimensional subspaces of X, it is, in a certain sense, a measure of the "thickness"
or "massivity" of A.
It is, of course, generally impossible to obtain d (A; X) and determine optimal
n
subspaces Xn for dn(A;X) (if they exist) for all A and X. Nonetheless, much effort
has been devoted to this task for specific choices of A and X. In this work we
survey numerous choices of A and X for which both dn(A;X) and Xn are in fact
obtained, or at least characterized. However in some surprisingly simple cases (see
for example Chapter VI) these quantities have not as yet been calculated, though
hardly through lack of effort.
It is also of considerable interest to determine the asymptotic behavior of
dn(A; X) as n i 00, since typically an optimal Xn cannot be explicitly obtained or
if it can, too much computational effort is involved in its determination. In many
cases very simple n-dimensional subspaces may approximate A in an asymp
totically optimal manner. Thus the n-width, in providing a lower threshold on the
degree of approximation of A by n-dimensional subspaces, tells us how well a
given n-dimensional subspace (e.g., algebraic or trigonometric polynomials,
splines with fixed knots, etc ...) approximates A relative to the theoretical lower
bound. On this basis it is then possible to judge whether the additional time, effort,
and perhaps money, involved in using better but more complicated subspaces is,
in fact, justified.
Kolmogorov, aside from simply defining d (A; X), also computed this quan
n
tity in two particular instances. The first of these we give here; the second is
deferred to Chapter IV.
Let L2 = L2 [0,211:] denote the usual space of square integrable functions on
[0,211:] with norm
)1/2
21<
11111 = ( 1/211:! II(xWdx .
For any given positive integer r, let Wj') denote the (Sobolev) space of2 11:-periodic,
real-valued, (r - I)-times differentiable functions whose (r - l)st derivative is ab-
I. Introduction 3
solutely continuous and whose rth derivative is in L2. Thus
wt) = {f:j<,-1) abs. cont., j<')e L2, j<i)(0) = j<i)(2 n), i = 0, 1, ... , r - 1}.
Set
Jj~) = {f:f e Wi'), II j<') II ~ 1},
and let us consider dn(Jj~);L2).
Theorem 1 (Kolmogorov [1936]).
Furthermore, an optimal subspace for d2n -1 (Jj~); L2) (and hence for d2n(Jj~); L2» is
T,,-1 = span {1, sin x, cos x, ... , sin(n - 1) x, cos(n - 1) x}
i.e., trigonometric polynomials of degree less than or equal to n - 1.
Before discussing the proof of Theorem 1, we remark that T" _ is not the only
1
optimal subspace for d2n(Jj~);L2). In Chapter IV, we construct an additional op
timal subspace. In what follows we give a sketch of the proof of Theorem 1. The
proof may easily be made rigorous and this is done in Chapter IV.
"Proof". Since every constant function is in Jj~), and do(Jj~);L2) =
sup {II f II:f e Jj~)}, it follows that do(Jj~);L2) = 00. For this very same reason it
may be easily seen that if E(Jj~);Xn) < 00, then it is necessary that the constant
function be contained in Xn•
We first obtain an upper bound on d2n-1 (Jj~); L2) by showing that
E(Jj~); T,,-1) ~ n-'. By definition
E (Jj~); T,,-1) = sup inf II f - til.
jeB'!) teT.-l
Assume that f is ofthe form f (x) = ao + L<Xl ak cos k x + bk sin k x. The condition
f e Jj~) is equivalent to k= 1
1/2 L<Xl (la,,12+ Ib,,12) p' ~ 1.
k=1
Since we wish to bound E(Jj~); T,,-1) from above, it suffices to set t(x) =
n-1
ao + L ak cos kx + b" sin kx in the above expression (this is, in fact, the best
k=1
approximation), so that