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Volume 5

Trigonometrical series

Antoni Zygmund

Warszawa-Lwów 1935


CHAPITRE I Trigonometrical series and Fourier serie 

1.1. Definitions.
1.2. Abel's transformation.
1.3. Orthogonal systems of functions. Fourier series.
1.4. The trigonometrical system.
1.5. Completness of the trigonometrical system.
1.6. Bessel's inequality. Farseval's relation.
1.7. Remarks on series and integrals.
1.8. Miscellaneous theorems and examples. 

CHAPITRE II Fourier coefficients. Tests for the convergence of 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 dé la Val-lée-Poussin, Young, and Hardy and Littlewood.
2.9. Miscellaneous theorems and examples. 

CHAPITRE III Summability oi Fourier series 

3.1. Toeplitz matrices. Abel and Cesaro 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 Cesaro 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.  

CHAPITRE IV Classes of functions and Fourier series 

4.1. Inequalities
4.2. Mean convergence. The Riesz-Fischer theorem.
4.3. Classes B, C, S, and Ltri of functions.
4.4. Parseval's relations.
4.5. Linear operations.
4.6. Transformations of Fourier series.
4.7. Miscellaneous theorems and examples. 

CHAPITRE 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. La-cunary series.
5.5. Rademacher's series.
5.6. Applications of Ra-demacher's functions.
5.7. Miscellaneous theorems and examples. 

CHAPITRE 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 Levy.
6.6. Miscellaneous theorems and examples. 

CHAPITRE VII Conjugate series and complex methods in the theory of Fourier series 

7.1. Summability of conjugate series.
7.2. Conjugate series and Fourier series.
7.3. Mean convergence of Fourier series.
7.4. Priva-loff's theorem.
7.5. Power series of bounded variation.
7.6. Miscellaneous theorems and examples. 

CHAPITRE VIII Divergence of 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. Kol-mogoroffs example.
8.5. Gibbs's phenomenon.
8.6. Theorems of Rogosinski.
8.7. Cramer's theorem.
8.8. Miscellaneous theorems and examples. 

CHAPITRE 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'a convexity theorems.
9.3. Proof of F. Riesz's theorem.
9.4. Theorems of Paley.
9.5. Theorems of Hardy and Little-wood.
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. 

CHAPITRE 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.
10.4. Summability C of Fourier series.
10.5. Miscellaneous theorems and examples. 

CHAPITRE XI Riemann's theory of trigonometrical series 

11.1. The Cantor-Lebesgue theorem and its generalization.
11.2. Rie-mann's and Fatou's theorems.
11.3. Theorems of uniqueness.
11.4. The principle of localization. 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. 

CHAPITRE XII Fourier's integral 

12.1. Fourier's single integral.
12.2. Fourier's repeated integral.
12 3. Suramability of integrals.
12.4. Fourier transforms. 

Materiały redakcyjne  

Preface, errata, terminological index, bibliography, table of contents 

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