Theory of Approximation

by Achieser, N. I.

• ISBN13: 9780486671291
• ISBN10: 0486671291
• Edition: Reprint
• Format: Paperback
• Copyright: 1992-06-01
• Publisher: Dover Pubns

Summary

A pioneer of many modern developments in approximation theory, Achieser begins this text with approximation problems in linear normalized spaces and the ideas of P. L. Tchebysheff. He then examines the elements of harmonic analysis, integral transcendental functions of the exponential type, Wiener's theorem on approximation, more. Includes an extensive section of problems and applications.

Table of Contents

Approximation Problems in Linear Normalized Spaces
Formulation of the Principal Problem in the Theory of Approximation	p. 1
The Concept of Metric Space	p. 1
The Concept of Linear Normalized Space	p. 2
Examples of Linear Normalized Spaces	p. 3
The Inequalities of Holder and Minkowski	p. 4
Additional Examples of Linear Normalized Spaces	p. 7
Hilbert Space	p. 8
The Fundamental Theorem of Approximation Theory in Linear Normalized Spaces	p. 10
Strictly Normalized Spaces	p. 11
An Example of Approximation in the Space L[superscript p]	p. 12
Geometric Interpretation	p. 13
Separable and Complete Spaces	p. 14
Approximation Theorems in Hilbert Space	p. 15
An Example of Approximation in Hilbert Space	p. 19
More About the Approximation Problem in Hilbert Space	p. 21
Orthonormalized Vector Systems in Hilbert Space	p. 22
Orthogonalization of Vector Systems	p. 23
Infinite Orthonormalized Systems	p. 25
An Example of a Non-Separable System	p. 29
Weierstrass' First Theorem	p. 29
Weierstrass' Second Theorem	p. 32
The Separability of the Space C	p. 33
The Separability of the Space L[superscript p]	p. 34
Generalization of Weierstrass' Theorem to the Space L[superscript p]	p. 37
The Completeness of the Space L[superscript p]	p. 38
Examples of Complete Orthonormalized Systems in L[superscript 2]	p. 40
Muntz's Theorem	p. 43
The Concept of the Linear Functional	p. 46
F. Riesz's Theorem	p. 47
A Criterion for the Closure of a Set of Vectors in Linear Normalized Spaces	p. 49
P. L. Tchebysheff's Domain of Ideas
Statement of the Problem	p. 51
A Generalization of the Theorem of de la Vallee-Poussin	p. 52
The Existence Theorem	p. 53
Tchebysheff's Theorem	p. 55
A Special Case of Tchebysheff's Theorem	p. 57
The Tchebysheff Polynomials of Least Deviation from Zero	p. 57
A Further Example of P. Tchebysheff's Theorem	p. 58
An Example for the Application of the General Theorem of de la Vallee-Poussin	p. 60
An Example for the Application of P. L. Tchebysheff's General Theorem	p. 62
The Passage to Periodic Functions	p. 64
An Example of Approximating with the Aid of Periodic Functions	p. 66
The Weierstrass Function	p. 66
Haar's Problem	p. 67
Proof of the Necessity of Haar's Condition	p. 68
Proof of the Sufficiency of Haar's Condition	p. 69
An Example Related to Haar's Problem	p. 72
P. L. Tchebysheff's Systems of Functions	p. 73
Generalization of P. L. Tchebysheff's Theorem	p. 74
On a Question Pertaining to the Approximation of a Continuous Function in the Space L	p. 76
A. A. Markoff's Theorem	p. 82
Special Cases of the Theorem of A. A. Markoff	p. 85
Elements of Harmonic Analysis
The Simplest Properties of Fourier Series	p. 89
Fourier Series for Functions of Bounded Variation	p. 93
The Parseval Equation for Fourier Series	p. 97
Examples of Fourier Series	p. 98
Trigonometric Integrals	p. 101
The Riemann-Lebesgue Theorem	p. 103
Plancherel's Theory	p. 104
Watson's Theorem	p. 106
Plancherel's Theorem	p. 108
Fejer's Theorem	p. 110
Integral-Operators of the Fejer Type	p. 113
The Theorem of Young and Hardy	p. 116
Examples of Kernels of the Fejer Type	p. 118
The Fourier Transformation of Integrable Functions	p. 120
The Faltung of two Functions	p. 122
V. A. Stekloff's Functions	p. 123
Multimonotonic Functions	p. 125
Conjugate Functions	p. 126
Certain Extremal Properties of Integral Transcendental Functions of the Exponential Type
Integral Functions of the Exponential Type	p. 130
The Borel Transformation	p. 132
The Theorem of Wiener and Paley	p. 134
Integral Functions of the Exponential Type which are Bounded along the Real Axis	p. 137
S. N. Bernstein's Inequality	p. 140
B. M. Levitan's Polynomials	p. 146
The Theorem of Fejer and Riesz. A Generalization of This Theorem	p. 152
A Criterion for the Representation of Continuous Functions as Fourier-Stieltjes Integrals	p. 154
Questions Regarding the Best Harmonic Approximation of Functions
Preliminary Remarks	p. 160
The Modulus of Continuity	p. 161
The Generalization to the Space L[superscript p] (p [greater than or equal] 1)	p. 162
An Example of Harmonic Approximation	p. 165
Some Estimates for Fourier Coefficients	p. 169
More about V. A. Stekloff's Functions	p. 173
Two Lemmas	p. 175
The Direct Problem of Harmonic Approximation	p. 176
A Criterion due to B. Sz.-Nagy	p. 183
The Best Approximation of Differentiable Functions	p. 187
Direct Observations Concerning Periodic Functions	p. 195
Jackson's Second Theorem	p. 199
The Generalized Fejer Method	p. 201
Berstein's Theorem	p. 206
Priwaloff's Theorem	p. 210
Generalizations of Bernstein's Theorems to the Space L[superscript p] (p [greater than or equal] 1)	p. 211
The Best Harmonic Approximation of Analytic Functions	p. 214
A Different Formulation of the Result of the Preceding Section	p. 218
The Converse of Bernstein's Theorem	p. 221
Wiener's Theorem on Approximation
Wiener's Problem	p. 224
The Necessity of Wiener's Condition	p. 224
Some Definitions and Notation	p. 225
Several Lemmas	p. 227
The Wiener-Levy Theorem	p. 230
Proof of the Sufficiency of Wiener's Condition	p. 233
Wiener's General Tauber Theorem	p. 234
Weakly Decreasing Functions	p. 235
Remarks on the Terminology	p. 237
Ikehara's Theorem	p. 238
Carleman's Tauber Theorem	p. 241
Various Addenda and Problems
Elementary Extremal Problems and Certain Closure Criteria	p. 243
Szego's Theorem and Some of Its Applications	p. 256
Further Examples of Closed Sequences of Functions	p. 267
The Caratheodory-Fejer Problem and Similar Problems	p. 270
Solotareff's Problems and Related Problems	p. 280
The Best Harmonic Approximation of the Simplest Analytic Functions	p. 289
Notes	p. 296
Index	p. 306
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