Description

This book will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis. 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. 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. 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. Introduction; About this book; 1. Trigonometric series and sets of uniqueness; 2. The algebra A of functions with absolutely convergent fourier series, pseudofunctions and pseudomeasures; 3. Symmetric perfect sets and the Salem-Zygmund theorem; 4. Classification of the complexity of U; 5. The Piatetski-Shapiro hierarchy of U-sets; 6. Decomposing U-sets into simpler sets; 7. The shrinking method, the theorem of Körner and Kaufman, and the solution to the Borel basis problem for U; 8. Extended uniqueness sets; 9. Characterizing Rajchman measures; 10. Sets of resolution and synthesis; List of problems; References; Symbols and Abbreviations; Index. "Of all the work that has been done in recent years on connections between descriptive set theory and analysis, the results contained in this book are the deepest and most significant." Mathematical Reviews