Trigonometrical series

Antoni Zygmund

GDML_Books, (1935), p.

Résumé

PREFACE................................... III ERRATA.................................... IV CHAPTER I. Trigonometrical series and Fourier series...... 1 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. CHAPTER II. Fourier coefficients. Tests for the convergence of Fourier series........... 14 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.......... 40 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. CHAPTER IV. Classes of functions and Fourier series.... 64 4.1. Inequalities 4.2. Mean convergence. The Riesz-Fischer theorem. 4.3. Classes B, C, S, and $L_φ$ of functions. 4.4. 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....... 108 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.......... 131 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. CHAPTER VII. Conjugate series and complex methods in the theory of Fourier series........... 145 7.1. Summability 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 of Fourier series. Gibbs’s phenomenon...................... 167 8.1. Continuous functions with divergent Fourier series. 8.2. A theorem of Faber and Lebesgue. 8.3. Lebesgue’s constants. 8.4. Kolmogoroffs example. 8.5. Gibbs’s phenomenon. 8.6. Theorems of Rogosinski. 8.7. Cramer’s theorem. 8.8. Miscellaneous theorems and examples. CHAPTER IX. Further theorems on Fourier coefficients. Integration of fractional order............. 189 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 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............. 237 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. CHAPTER XI. Riemann’s theory of trigonometrical series....................... 267 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 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. CHAPTER XII. Fourier’s integral....................... 306 12.1. Fourier’s single integral. 12.2. Fourier’s repeated integral. 12 3. Summability of integrals. 12.4. Fourier transforms. TERMINOLOGICAL INDEX, NOTATIONS........................ 320 BIBLIOGRAPHY.......................... 321

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