| Preface | p. xi |
| List of Symbols | p. xiv |
| Trigonometric Series and Fourier Series. Auxiliary Results | |
| Trigonometric series | p. 1 |
| Summation by parts | p. 3 |
| Orthogonal series | p. 5 |
| The trigonometric system | p. 6 |
| Fourier-Stieltjes series | p. 10 |
| Completeness of the trigonometric system | p. 11 |
| Bessel's inequality and Parseval's formula | p. 12 |
| Remarks on series and integrals | p. 14 |
| Inequalities | p. 16 |
| Convex functions | p. 21 |
| Convergence in L[superscript r] | p. 26 |
| Sets of the first and second categories | p. 28 |
| Rearrangements of functions. Maximal theorems of Hardy and Littlewood | p. 29 |
| Miscellaneous theorems and examples | p. 34 |
| Fourier Coefficients. Elementary Theorems on The Convergence of S[f] and S[f] | |
| Formal operations on S[f] | p. 35 |
| Differentiation and integration of S[f] | p. 40 |
| Modulus of continuity. Smooth functions | p. 42 |
| Order of magnitude of Fourier coefficients | p. 45 |
| Formulae for partial sums of S[f] and S[f] | p. 49 |
| The Dini test and the principle of localization | p. 52 |
| Some more formulae for partial sums | p. 55 |
| The Dirichlet-Jordan test | p. 57 |
| Gibbs's phenomenon | p. 61 |
| The Dini-Lipschitz test | p. 62 |
| Lebesgue's test | p. 65 |
| Lebesgue constants | p. 67 |
| Poisson's summation formula | p. 68 |
| Miscellaneous theorems and examples | p. 70 |
| Summability of Fourier Series | |
| Summability of numerical series | p. 74 |
| General remarks about the summability of S[f] and S[f] | p. 84 |
| Summability of S[f] and S[f] by the method of the first arithmetic mean | p. 88 |
| Convergence factors | p. 93 |
| Summability (C, [alpha]) | p. 94 |
| Abel summability | p. 96 |
| Abel summability (cont.) | p. 99 |
| Summability of S[dF] and S[dF] | p. 105 |
| Fourier series at simple discontinuities | p. 106 |
| Fourier sine series | p. 109 |
| Gibbs's phenomenon for the method (C, [alpha]) | p. 110 |
| Theorems of Rogosinski | p. 112 |
| Approximation to functions by trigonometric polynomials | p. 114 |
| Miscellaneous theorems and examples | p. 124 |
| Classes of Functions and Fourier Series | |
| The class L[superscript 2] | p. 127 |
| A theorem of Marcinkiewicz | p. 129 |
| Existence of the conjugate function | p. 131 |
| Classes of functions and (c, 1) means of Fourier series | p. 134 |
| Classes of functions and (C, 1) means of Fourier series (cont.) | p. 143 |
| Classes of functions and Abel means of Fourier series | p. 149 |
| Majorants for the Abel and Cesaro means of s[f] | p. 154 |
| Parseval's formula | p. 157 |
| Linear operations | p. 162 |
| Classes L*[subscript Phi] | p. 170 |
| Conversion factors for classes of Fourier series | p. 175 |
| Miscellaneous theorems and examples | p. 179 |
| Special Trigonometric Series | |
| Series with coefficients tending montonically to zero | p. 182 |
| The order of magnitude of functions represented by series with monotone coefficients | p. 186 |
| A class of Fourier-Stieltjes series | p. 194 |
| The series [Sigma]n[superscript -1/2-alpha] e[superscript icn log n] e[superscript inx] | p. 197 |
| The series [Sigma]v[superscript -beta] e[superscript iv superscript alpha] e[superscript ivx] | p. 200 |
| Lacunary series | p. 202 |
| Riesz products | p. 208 |
| Rademacher series and their applications | p. 212 |
| Series with 'small' gaps | p. 222 |
| A power series of Salem | p. 225 |
| Miscellaneous theorems and examples | p. 228 |
| The Absolute Convergence of Trigonometric Series | |
| General series | p. 232 |
| Sets N | p. 235 |
| The absolute convergence of Fourier series | p. 240 |
| Inequalities for polynomials | p. 244 |
| Theorems of Wiener and Levy | p. 245 |
| The absolute convergence of lacunary series | p. 247 |
| Miscellaneous theorems and examples | p. 250 |
| Complex Methods in Fourier Series | |
| Existence of conjugate functions | p. 252 |
| The Fourier character of conjugate series | p. 253 |
| Applications of Green's formula | p. 260 |
| Integrability B | p. 262 |
| Lipschitz conditions | p. 263 |
| Mean convergence of S[f] and S[f] | p. 266 |
| Classes H[superscript p] and N | p. 271 |
| Power series of bounded variation | p. 285 |
| Cauchy's integral | p. 288 |
| Conformal mapping | p. 289 |
| Miscellaneous theorems and examples | p. 295 |
| Divergence of Fourier Series | |
| Divergence of Fourier series of continuous functions | p. 298 |
| Further examples of divergent Fourier series | p. 302 |
| Examples of Fourier series divergent almost everywhere | p. 305 |
| An everywhere divergent Fourier series | p. 310 |
| Miscellaneous theorems and examples | p. 314 |
| Riemann's Theory of Trigonometric Series | |
| General remarks. The Cantor-Lebesgue theorem | p. 316 |
| Formal integration of series | p. 319 |
| Uniqueness of the representation by trigonometric series | p. 325 |
| The principle of localization. Formal multiplication of trigonometric series | p. 334 |
| Formal multiplication of trigonometric series (cont.) | p. 337 |
| Sets of uniqueness and sets of multiplicity | p. 344 |
| Uniqueness of summable trigonometric series | p. 352 |
| Uniqueness of summable trigonometric series (cont.) | p. 356 |
| Localization for series with coefficients not tending to zero | p. 363 |
| Miscellaneous theorems and examples | p. 370 |
| Notes | p. 375 |
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