# Book:Antoni Zygmund/Trigonometrical Series

## Antoni Zygmund: Trigonometrical Series

Published $1935$, Dover Publications.

### 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