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1985 | Buch

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n-Widths in Approximation Theory

verfasst von: Allan Pinkus

Verlag: Springer Berlin Heidelberg

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Professional Book Archive

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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 1 Chapter I. Introduction . . . . . . . . Chapter II. Basic Properties of n-Widths . 9 1. Properties of d • • • • • • • • • • 9 n 15 2. Existence of Optimal Subspaces for d • n n 17 3. Properties of d • • • • • • 20 4. Properties of b • • • • • • n 5. Inequalities Between n-Widths 22 n 6. Duality Between d and d • • 27 n 7. n-Widths of Mappings of the Unit Ball 29 8. Some Relationships Between dn(T), dn(T) and bn(T) . 32 37 Notes and References . . . . . . . . . . . . . .

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction

Abstract

In this short chapter we introduce the subject matter. We hope that this introduction whets the reader’s appetite for the more systematic treatment which will follow.

Allan Pinkus

Chapter II. Basic Properties of n-Widths

Abstract

Let X be a normed linear space and X_n an n-dimensional subspace of X. For each x ∈ X, E(x;X_n) is the distance of the n-dimensional subspace X_n from x, defined by

$$E(x;{X_n}) = \inf \left\{ {\left\| {x - y} \right\|:y \in {X_n}} \right\}.$$

Allan Pinkus

Chapter III. Tchebycheff Systems and Total Positivity

Abstract

This chapter is not concerned with n-widths but is a discussion of Tchebycheff (T-) systems (often written Chebyshev) and total positivity which are important tools in the exact determination of n-widths and in the identification of optimal subspaces for many of the examples considered in this work. While we do assume that the reader is familiar with some of the basic results of approximation theory and related matters, we cannot assume that the reader also has a familiarity with T-systems and total positivity. It transpires that the theory of T-systems and total positivity is intimately connected with the problem of zero counting and oscillations of functions and, as such, is basic to the study of L^∞ and L¹ approximations, as shall be evident from results of succeeding chapters. Perhaps surprisingly it is also important in the L² theory of n-widths.

Allan Pinkus

Chapter IV. n-Widths in Hilbert Spaces

Abstract

The organization of this chapter is as follows. The most general theorem concerning n-widths in Hilbert spaces is the main content of Section 2. Let T be a compact operator mapping H₁ to H₂, where both H₁ and H₂ are Hilbert spaces. Then the Kolmogorov, linear, Gel’fand, and Bernstein n-width of the set

$$A = \left\{ {T\Phi :{{\left\| \Phi \right\|}_{{H_1}}}\underline{\underline < } 1} \right\}$$

as a subset of H₂, is simply the (n + 1)st singular value of T and optimal subspaces are easily constructed in terms of eigenvector subspaces. In Section 3, we consider variations on this problem and many examples. The n-widths and optimal subspaces are easily identified if T is given by convolution against a fixed periodic function. This is discussed in Section 4. Sections 5 and 6 are different in nature. In those sections we consider integral operators whose kernels are either totally positive or cyclic variation diminishing (and variants thereof). In these cases we are able to determine additional optimal subspaces of an elementary form.

Allan Pinkus

Chapter V. Exact n-Widths of Integral Operators

Abstract

Let $K(x,y) \in C([0,1]) \times [0,1])$ and set

$${K_p} = \left\{ {K\,h\,\left( x \right):K\,h\,\left( x \right) = \int\limits_0^1 {K\,(x,y)\,h\,(y)\,d\,y,{{\left\| h \right\|}_p}} \underline{\underline < } \,1} \right\}$$

where

$${\left\| h \right\|_p} = \left\{ {_{ess\sup \,\left\{ {\left| {h\left( y \right)} \right|:\,0\underline{\underline < } \,y\,\underline{\underline < } \,1} \right\},\,\,\,\,\,p = \infty }^{{{\left( {\int\limits_0^1 {{{\left| {h\left( y \right)} \right|}^p}dy} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\underline{\underline < } p < \infty }} \right.$$

In Section 2 of this chapter we determine the n-widths (d_n, dⁿ and δ _n) of K_p in L^q for p = ∞, 1 ≦ q ≦ ∞, and 1 ≦ p ≦ ∞, q = 1, where K is a nondegenerate totally positive kernel. (Such kernels were defined in Section 5 of Chapter IV and are redefined in Section 2.) We prove that all three n-widths considered are equal (the common value depending on p and q) and that there exists a set of n distinct points in {E_i} _i=1 ⁿ in (0,1) for which X_n = span {K(•,E_i)} _i=1 ⁿ is optimal for d_n(K_p;L^q), with the {E_i} _i=1 ⁿ dependent on p, q. Furthermore, there exists an additional set {η_i} _i=1 ⁿ of n distinct points in (0,1) (again dependent on p, q) for which interpolation from X_n to K h(x) at the {η_i} _i=1 ⁿ is optimal for δ_n (K_p;L^q). (An analogous statement holds for dⁿ(K_p;L^q).)

Allan Pinkus

Chapter VI. Matrices and n-Widths

Abstract

Let $A = ({a_{ij}})_i,j=1^M$ be an M x M real matrix. (We shall deal with real square matrices for notational convenience.) For x ∈ℝ^M, set

$${\left\| X \right\|_p} = \left\{ {_{\mathop {\max }\limits_{i = 1,...,M} \left| {{X_i}} \right|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p = \infty .}^{{{\left( {\sum\limits_{i = 1}^M {{{\left| {{X_i}} \right|}^p}} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}},\,\,\,\,\,1\underline{\underline < } \,p\, < \,\infty ,}} \right.$$

Allan Pinkus

Chapter VII. Asymptotic Estimates for n-Widths of Sobolev Spaces

Abstract

Let W _p ^(r) [0,1] = W _p ^(r) = {f:f ∈ C^r ^-1 [0,1],f^(r-1) abs. cont., f^(r) ∈ L^p [0,1]}, and set B _p ^(r) = {f: f ∈ W _p ^(r) , ∥ f^(r)∥_p ≦ 1}. We are here concerned with the asymptotic behaviour of the values d_n(B _p ^(r) ;L^q), dⁿ(B _p ^(r) ; L^q) and δ _n (B _p ^(r) ;L^q) for p,q, ∈[1, ∞] as n grows. It is only recently that this problem has been solved for all p, q ∈ [1, ∞] (and almost all r). We first state the major result proven in this chapter.

Allan Pinkus

Chapter VIII. n-Widths of Analytic Functions

Abstract

In this chapter we consider n-widths of various classes of analytic functions. Our concern is with exact n-widths and the identification of optimal subspaces. Asymptotic estimates are relegated to the “Notes and References” at the end of the chapter.

Allan Pinkus

Backmatter

Titel: n-Widths in Approximation Theory
verfasst von: Allan Pinkus
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-69894-1
Print ISBN: 978-3-642-69896-5
DOI: https://doi.org/10.1007/978-3-642-69894-1

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction

Chapter II. Basic Properties of n-Widths

Chapter III. Tchebycheff Systems and Total Positivity

Chapter IV. n-Widths in Hilbert Spaces

Chapter V. Exact n-Widths of Integral Operators

Chapter VI. Matrices and n-Widths

Chapter VII. Asymptotic Estimates for n-Widths of Sobolev Spaces

Chapter VIII. n-Widths of Analytic Functions

Backmatter