| Description |
1 online resource (367 pages) : illustrations |
| Series |
London Mathematical Society lecture note series ; 128 |
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London Mathematical Society lecture note series ; 128.
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| Contents |
Cover; Title; Copyright; Dedication; Preface; Contents; INTRODUCTION; ABOUT THIS BOOK; CHAPTER I. TRIGONOMETRIC SERIES AND SETS OF UNIQUENESS; 1. Trigonometric and Fourier Series; 2. The problem of uniqueness; 3. The Riemann theory and the Cantor Uniqueness Theorem; 4. The Rajchman multiplication theory. Examples of perfect sets of uniqueness; 5. Countable unions of closed sets of uniqueness; 6. Four classical problems; CHAPTER II. THE ALGEBRA A OF FUNCTIONS WITH ABSOLUTELY CONVERGENT FOURIER SERIES, PSEUDOFUNCTIONS AND PSEUDOMEASURES; 1. The spaces PF, A and PM; 2. Some basic facts about A |
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3. Supports of pseudomeasures4. Description of closed 'U-sets in terms of pseudofunctions; 5. Rajchman measures and extended uniqueness sets; CHAPTER III. SYMMETRIC PERFECT SETS AND THE SALEM-ZYGMUND THEOREM; 1. H(n)-sets; 2. Pisot numbers; 3. Symmetric and homogeneous perfect sets; 4. The Salem-Zygmund Theorem; CHAPTER IV. CLASSIFICATION OF THE COMPLEXITY OF U; 1. Some descriptive set theory; 2. The theorem of Solovay and Kaufman; 3. On cr-ideals of closed sets in compact, metrizable spaces; CHAPTER V. THE PIATETSKI-SHAPIRO HIERARCHY OF U-SETS; 1. IlJ-ranks on Il{ sets |
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2. Ranks for subspaces of Banach spaces3. The tree-rank and the R-rank; 4. The Piatetski-Shapiro rank on U; 5. The class U' of uniqueness sets of rank 1; CHAPTER VI. DECOMPOSING U-SETS INTO SIMPLER SETS; 1. Borel bases for a-ideals of closed sets; 2. The class l̂ and the decomposition theorem of Piatetski-Shapiro; 3. The Borel Basis Problem for U and relations between U, Ux and Uo; CHAPTER VII. THE SHRINKING METHOD, THE THEOREM OF KORNER AND KAUFMAN, AND THE SOLUTION TO THE BOREL BASIS PROBLEM FOR U; 1. Sets of interior uniqueness; 2. Approximating M-sets by Ha-sets |
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3. Helson sets of multiplicity4. The solution to the Borel Basis Problem; CHAPTER VIII. EXTENDED UNIQUENESS SETS; 1. The class U'o; 2. The existence of a Borel basis for Uo and its associated rank; 3. The solution to the Category Problem, and other applications; 4. The class Vl revisited; CHAPTER IX. CHARACTERIZING RAJCHMAN MEASURES; 1. A theorem of Mokobodzki in measure theory; 2. W-sets and Lyons* characterization of Rajchman measures; CHAPTER X. SETS OF RESOLUTION AND SYNTHESIS; 1. Sets of resolution; 2. Sets of synthesis; LIST OF PROBLEMS; REFERENCES; SYMBOLS AND ABBREVIATIONS; INDEX |
| Summary |
The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis |
| Bibliography |
Includes bibliographical references (pages 353-358) and index |
| Notes |
Print version record |
| Subject |
Descriptive set theory.
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Fourier series.
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MATHEMATICS -- Infinity.
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Descriptive set theory
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Fourier series
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Deskriptive Mengenlehre
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Eindeutigkeit
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Trigonometrische Reihe
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Ensembles, Théorie descriptive des.
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Fourier, Séries de.
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Théorèmes d'unicité.
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| Form |
Electronic book
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| Author |
Louveau, Alain
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| ISBN |
9781107361492 |
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1107361494 |
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