Alexits Convergence Problems of Orthogonal Series

1. Auflage 2014
ISBN: 978-1-4832-2277-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 20, 362 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Convergence Problems of Orthogonal Series deals with the theory of convergence and summation of the general orthogonal series in relation to the general theory and classical expansions. The book reviews orthogonality, orthogonalization, series of orthogonal functions, complete orthogonal systems, and the Riesz-Fisher theorem. The text examines Jacobi polynomials, Haar's orthogonal system, and relations to the theory of probability using Rademacher's and Walsh's orthogonal systems. The book also investigates the convergence behavior of orthogonal series by methods belonging to the general theory of series. The text explains some Tauberian theorems and the classical Abel transform of the partial sums of a series which the investigator can use in the theory of orthogonal series. The book examines the importance of the Lebesgue functions for convergence problems, the generalization of the Walsh series, the order of magnitude of the Lebesgue functions, and the Lebesgue functions of the Cesaro summation. The text also deals with classical convergence problems in which general orthogonal series have limited significance as orthogonal expansions react upon the structural properties of the expanded function. This reaction happens under special assumptions concerning the orthogonal system in whose functions the expansion proceeds. The book can prove beneficial to mathematicians, students, or professor of calculus and advanced mathematics.

Autoren/Hrsg.

Alexits, G.

Inhaltsverzeichnis

1;Front Cover;1
2;Convergence Problems of Orthogonal Series;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE TO THE ENGLISHEDITION;8
6;CHAPTER I. FUNDAMENTAL IDEAS. EXAMPLES OF SERIES OF ORTHOGONAL FUNCTIONS;12
6.1;§ 1. Orthogonality, orthogonalization, series of orthogonal functions;12
6.2;§ 2. The space L2µ. The Riesz–Fischer theorem. Complete orthogonal systems;20
6.3;§ 3. Orthogonal polynomials;32
6.4;§ 4. The Jacobi polynomials;41
6.5;§ 5. Bounds for general orthonormal systems and orthonormal systems of polynomials;48
6.6;§ 6. Haar's orthogonal system;57
6.7;§ 7. Rademacher's and Walsh's orthogonal systems. Relations to the theory of probability;62
7;CHAPTER II. INVESTIGATION OF THE CONVERGENCE BEHAVIOUR OF ORTHOGONAL SERIES BY METHODS BELONGING TO THE GENERAL THEORY OF SERIES;74
7.1;§ 1. General summation processes;76
7.2;§ 2. The Abel transform. Some Tauberian theorems;82
7.3;§ 3. The fundamental theorem concerning the convergence of orthogonal series;90
7.4;§ 4. Everywhere divergent orthogonal series;98
7.5;§ 5. Convergence with every arrangement of the terms;113
7.6;§ 6. Generalities on the Cesàro summation of orthogonal series;120
7.7;§ 7. Convergence of subsequences and Cesàro summability;129
7.8;§ 8. Coefficient tests for the Cesàro summability of orthogonal series;136
7.9;§ 9. Abel non-summable orthogonal series with monotone coefficients;154
7.10;§ 10. Menchoff's summation theorem;163
7.11;§ 11. Orthogonal universal series;171
8;CHAPTER III. THE LEBESGUE FUNCTIONS;182
8.1;§ 1. The significance of the Lebesgue functions for convergence problems;182
8.2;§ 2. Multiplicatively orthogonal systems. Generalization of the Walsh series;196
8.3;§ 3. The Lebesgue functions of the Cesàro summation;207
8.4;§ 4. Summability of orthogonal series arising in terms of the functions of a polynomial-like system;216
8.5;§ 5. The order of magnitude of the Lebesgue functions;224
8.6;§ 6. Impossibility of the improvement of the coefficient tests;238
9;CHAPTER IV. CLASSICAL CONVERGENCE PROBLEMS;249
9.1;§ 1. Banach spaces. Functionals;250
9.2;§ 2. The singular integrals;262
9.3;§ 3. Convergence of the polynomial-like orthogonal expansions at points of continuity;274
9.4;§ 4. The convergence features of the singular integrals at the Lebesgue points;282
9.5;§ 5. Generalities on the degree of approximation;297
9.6;§ 6· Approximation properties of some orthogonal systems;305
9.7;§ 7. Structural convergence conditions;319
9.8;§ 8. Absolute convergence of the orthogonal series;337
10;BIBLIOGRAPHY;350
11;GENERAL INDEX;359