Book contents
- Frontmatter
- Foreword
- TRIGONOMETRIC SERIES
- Contents
- Preface
- List of Symbols
- CHAPTER I TRIGONOMETRIC SERIES AND FOURIER SERIES. AUXILIARY RESULTS
- CHAPTER II FOURIER COEFFICIENTS. ELEMENTARY THEOREMS ON THE CONVERGENCE OF S[f] AND S[f]
- CHAPTER III SUMMABILITY OF FOURIER SERIES
- CHAPTER IV CLASSES OF FUNCTIONS AND FOURIER SERIES
- CHAPTER V SPECIAL TRIGONOMETRIC SERIES
- CHAPTER VI THE ABSOLUTE CONVERGENCE OF TRIGONOMETRIC SERIES
- CHAPTER VII COMPLEX METHODS IN FOURIER SERIES
- CHAPTER VIII DIVERGENCE OF FOURIER SERIES
- CHAPTER IX RIEMANN'S THEORY OF TRIGONOMETRIC SERIES
- Notes
- TRIGONOMETRIC SERIES
- CONTENTS
- CHAPTER X TRIGONOMETRIC INTERPOLATION
- CHAPTER XI DIFFERENTIATION OF SERIES. GENERALIZED DERIVATIVES
- CHAPTER XII INTERPOLATION OF LINEAR OPERATIONS. MORE ABOUT FOURIER COEFFICIENTS
- CHAPTER XIII CONVERGENCE AND SUMMABILITY ALMOST EVERYWHERE
- CHAPTER XIV MORE ABOUT COMPLEX METHODS
- CHAPTER XV APPLICATIONS OF THE LITTLE WOOD-PALEY FUNCTION TO FOURIER SERIES
- CHAPTER XVI FOURIER INTEGRALS
- CHAPTER XVII A TOPIC IN MULTIPLE FOURIER SERIES
- Notes
- References
- Index
Preface
Published online by Cambridge University Press: 05 February 2015
- Frontmatter
- Foreword
- TRIGONOMETRIC SERIES
- Contents
- Preface
- List of Symbols
- CHAPTER I TRIGONOMETRIC SERIES AND FOURIER SERIES. AUXILIARY RESULTS
- CHAPTER II FOURIER COEFFICIENTS. ELEMENTARY THEOREMS ON THE CONVERGENCE OF S[f] AND S[f]
- CHAPTER III SUMMABILITY OF FOURIER SERIES
- CHAPTER IV CLASSES OF FUNCTIONS AND FOURIER SERIES
- CHAPTER V SPECIAL TRIGONOMETRIC SERIES
- CHAPTER VI THE ABSOLUTE CONVERGENCE OF TRIGONOMETRIC SERIES
- CHAPTER VII COMPLEX METHODS IN FOURIER SERIES
- CHAPTER VIII DIVERGENCE OF FOURIER SERIES
- CHAPTER IX RIEMANN'S THEORY OF TRIGONOMETRIC SERIES
- Notes
- TRIGONOMETRIC SERIES
- CONTENTS
- CHAPTER X TRIGONOMETRIC INTERPOLATION
- CHAPTER XI DIFFERENTIATION OF SERIES. GENERALIZED DERIVATIVES
- CHAPTER XII INTERPOLATION OF LINEAR OPERATIONS. MORE ABOUT FOURIER COEFFICIENTS
- CHAPTER XIII CONVERGENCE AND SUMMABILITY ALMOST EVERYWHERE
- CHAPTER XIV MORE ABOUT COMPLEX METHODS
- CHAPTER XV APPLICATIONS OF THE LITTLE WOOD-PALEY FUNCTION TO FOURIER SERIES
- CHAPTER XVI FOURIER INTEGRALS
- CHAPTER XVII A TOPIC IN MULTIPLE FOURIER SERIES
- Notes
- References
- Index
Summary
The first edition of this book was written almost twenty-five years ago. Since then the theory of trigonometric series has undergone considerable change. It has always been one of the central parts of Analysis, but now we see its notions and methods appearing, in abstract form, in distant fields like the theory of groups, algebra, theory of numbers. These abstract extensions are, however, not considered here and the subject of the second edition of this book is, as before, the classical theory of Fourier series, which may be described as the meeting ground of the Real and Complex Variables.
This theory has been a source of new ideas for analysts during the last two centuries, and is likely to be so in years to come. Many basic notions and results of the theory of functions have been obtained by mathematicians while working on trigonometric series. Conceivably these discoveries might have been made in different contexts, but in fact they came to life in connexion with the theory of trigonometric series. It was not accidental that the notion of function generally accepted now was first formulated in the celebrated memoir of Dirichlet (1837) dealing with the convergence of Fourier series; or that the definition of Riemann's integral in its general form appeared in Riemann's Habilitationsschrift devoted to trigonometric series; or that the theory of sets, one of the most important developments of nineteenth-century mathematics, was created by Cantor in his attempts to solve the problem of the sets of uniqueness for trigonometric series.
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- Trigonometric Series , pp. xxiii - xxvPublisher: Cambridge University PressPrint publication year: 2003