# Book:Antoni Zygmund/Trigonometrical Series

## Antoni Zygmund: *Trigonometrical Series*

Published $1935$, **Dover Publications**.

### Subject Matter

### Contents

- Preface

- CHAPTER I.
**Trigonometrical series and Fourier series**- 1.1. Definitions. - 1.2. Abel's transformation. - 1.3. Orthogonal systems of functions. Fourier series. - 1.4. The trigonometrical system - 1.5. Completeness of the trigonometrical system. - 1.6. Bessel's inequality. Parseval's relation. - 1.7. Remark: on series and integrals - 1.8. Miscellaneous theorems and examples.

- CHAPTER II.
**Fourier coefficients. Tests for the convergence on Fourier series**- 2.1. Operations on Fourier series. - 2.2. Modulus of continuity. Fourier coefficients. - 2.3. Formulae for partial sums. - 2.4. Dini's test. - 2.5. Theorems on localization. - 2.6. Functions of bounded variation. - 2.7. Tests of Lebesgue and Dini-Lipschitz. - 2.8. Tests of de la Vallée-Poussin, Young, and Hardy and Littlewood. - 2.9. Miscellaneous theorems and examples.

- CHAPTER III.
**Summability of Fourier series**- 3.1. Toeplitz matrices. Abel and Cesàro means. - 3.2. Fejér's theorem. - 3.3. Summability $(C, r)$ of Fourier series and conjugate series - 3.4. Abel's summability. - 3.5. The Cesàro summation of differentiated series. - 3.6. Fourier sine series. - 3.7. Convergence factors. - 3.8. Summability of Fourier-Stieltjes series. - 3.9. Miscellaneous theorems and examples.

- CHAPTER IV.
**Classes on functions and Fourier series**- 4.1. Inequalities. - 4 2. Mean convergence. The Riesz-Fischer theorem. - 4.3. Classes $B$, $C$, $S$, and $L_\varphi$ of functions. - 3.4. Parseval's relations. - 4.5. Linear operations. - 4.6 Transformations of Fourier series. - 4.7. Miscellaneous theorems and examples.

- CHAPTER V.
**Properties of some special series**- 5.1. Series with coefficients monotonically tending to $0$. - 5.2. Approximate expressions for such series. - 5.3. A power series. - 5.4. Lacunary series. - 5.5. Rademacher's series. - 5 6. Applications of Rademacher's functions. - 5.7. Miscellaneous theorems and examples.

- CHAPTER VI.
**The absolute convergence of trigonometrical series**- 6.1. The Lusin-Denjoy theorem. - 6.2. Fatou's theorems. - 6.3. The absolute convergence of Fourier series. - 6.4. Szidon's theorem on lacunary series. - 6.5. The theorems of Wiener and Lévy. - 6.6. Miscellaneous theorems and examples.

- CHAPTER VII.
**Conjugate series and complex methods in the theory of Fourier series**- 7.1. Suitability of conjugate series. - 7.2. Conjugate series and Fourier series. - 7.3. Mean convergence of Fourier series. - 7.4. Privaloff's theorem. - 7.5. Power series of bounded variation. - 7.6. Miscellaneous theorems and examples.

- CHAPTER VIII.
**Divergence on Fourier series. Gibbs's phenomenon**- 8.1. Continuous functions with divergent Fourier series. - 8.2. A theorem of Faber and Lebesgue. - 8.3. Lebesgue's constants. - 8.4. Kolmogoroff's example. - 8.5. Gibbs's phenomenon. - 8.6. Theorems of Rogosinski. - 8.7. Cramér's theorem. - 8.8. Miscellaneous theorems and examples.

- CHAPTER IX.
**Further theorems on Fourier coefficients. Integration of fractional order**- 9.1. Remarks on the theorems of Hausdorff-Young and F. Riesz. - 9.2. M. Riesz's convexity theorems. - 9.3. Proof of F. Riesz's theorem. - 9.4. Theorems of Paley. - 9.5. Theorems of Hardy and Littlewood. - 9.6. Banach's theorems on lacunary coefficients. - 9.7. Wiener's theorem on functions of bounded variation. - 9.8. Integrals of fractional order. - 9.9. Miscellaneous theorems and examples.

- CHAPTER X.
**Further theorems on the summability and convergence of Fourier series**- 10.1. An extension of Fejér's theorem. - 10.2. Maximal theorems of Hardy and Littlewood. - 10.3. Partial sums of $\mathfrak S [f]$ for $f \in L^2$. - 10.4. Summability $C$ of Fourier series. - 10.5. Miscellaneous theorems and examples.

- CHAPTER XI.
**Riemannian theory of trigonometrical series**- 11.1. The Cantor-Lebesgue theorem and its generalization. - 11.2. Riemann's and Fatou's theorems. - 11.3. Theorems of uniqueness. - 11.4. The principle of localisation. Rajchman's theory of formal multiplication. - 11.5. Sets of uniqueness and sets of multiplicity. - 11.6. Uniqueness in the case of summable series. - 11.7. Miscellaneous theorems and examples.

- CHAPTER XII.
**Fourier's integral**- 12.1. Fourier's single integral. - 12.2. Fourier's repeated integral. - 12.3. Summability of integrals. - 12.4. Fourier transforms.

- TERMINOLOGICAL INDEX, NOTATIONS

- BIBLIOGRAPHY