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Descriptive set theory and the structure of sets of uniqueness /
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, mea...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge [Cambridgeshire] ; New York :
Cambridge University Press,
1987.
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| Colección: | London Mathematical Society lecture note series ;
128. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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
- 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
- 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 l^ 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
- 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