## Trigonometric Series, Volume 1Professor Zygmund's Trigonometric Series, first published in Warsaw in 1935, established itself as a classic. It presented a concise account of the main results then known, but on a scale that limited the amount of detailed discussion possible. A greatly enlarged second edition (Cambridge, 1959) published in two volumes took full account of developments in trigonometric series, Fourier series, and related branches of pure mathematics since the publication of the original edition. These two volumes, bound together with a foreword from Robert Fefferman, outline the significance of this text. Volume I, containing the completely re-written material of the original work, deals with trigonometric series and Fourier series. Volume II provides much material previously unpublished in book form. |

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### Contents

CHAPTER | 1 |

Summation by parts | 3 |

3 Orthogonal series | 5 |

The trigonometric system | 6 |

FourierStieltjes series | 10 |

6 Completeness of the trigonometric system | 11 |

Bessels inequality and Parsevals formula | 12 |

Remarks on series and integrals | 14 |

Lacunary series | 202 |

7 Riesz products | 208 |

Rademacher series and their applications | 212 |

Series with small gaps | 222 |

A power series of Salem | 225 |

Miscellaneous theorems and examples | 228 |

CHAPTER VI | 232 |

Sets N | 235 |

9 Inequalities | 16 |

Convex functions | 21 |

11 Convergence in Lr | 26 |

Sets of the first and second categories | 28 |

Rearrangements of functions Maximal theorems of Hardy and Littlewood | 29 |

Miscellaneous theorems and examples | 34 |

CHAPTER II | 35 |

Differentiation and integration of Sf | 40 |

Modulus of continuity Smooth functions | 42 |

Order of magnitude of Fourier coefficients | 45 |

Formulae for partial sums of S and S | 51 |

The Dini test and the principle of localization | 52 |

7 Some more formulae for partial sums | 55 |

The DirichletJordan test | 57 |

9 Gibbss phenomenon page | 61 |

The DiniLipschitz test | 62 |

11 Lebesgues test | 65 |

Lebesgue constants | 67 |

Poissons summation formula | 68 |

Miscellaneous theorems and examples | 70 |

CHAPTER III | 74 |

2 General remarks about the summability of S and S | 84 |

Summability of Sf and S by the method of the first arithmetic mean | 88 |

4 Convergence factors | 93 |

5 Summability C a | 94 |

Abel summability | 96 |

7 Abel summability cont | 99 |

8 Summability of SdF and SdF | 105 |

9 Fourier series at simple discontinuities | 106 |

10 Fourier sine series | 109 |

Gibbss phenomenon for the method C a | 110 |

12 Theorems of Rogosinski | 112 |

Approximation to functions by trigonometric polynomials | 114 |

Miscellaneous theorems and examples | 124 |

CHAPTER IV | 127 |

2 A theorem of Marcinkiewicz | 129 |

3 Existence of the conjugate function | 131 |

4 Classes of functions and C 1 means of Fourier series | 136 |

5 Classes of functions and C 1 means of Fourier series cont | 143 |

Classes of functions and Abel means of Fourier series | 149 |

7 Majorants for the Abel and Cesaro means of S page | 154 |

8 Parsevals formula | 157 |

9 Linear operations | 162 |

10 Classes LJ | 170 |

11 Conversion factors for classes of Fourier series | 175 |

Miscellaneous theorems and examples | 179 |

SPECIAL TRIGONOMETRIC SERIES 1 Series with coefficients tending monotonically to zero | 182 |

The order of magnitude of functions represented by series with monotone coefficients | 186 |

3 A class of FourierStieltjes series | 194 |

4 The series Zreiaeicnlon ei | 197 |

5 The series Svee11 | 200 |

The absolute convergence of Fourier series | 240 |

Inequalities for polynomials | 244 |

Theorems of Wiener and Levy | 245 |

The absolute convergence of lacunary series | 247 |

Miscellaneous theorems and examples | 250 |

COMPLEX METHODS IN FOURIER SERIES 1 Existence of conjugate functions page | 252 |

2 The Fourier character of conjugate series | 253 |

Applications of Greens formula | 260 |

Integrability B | 262 |

5 Lipschitz conditions | 263 |

Mean convergence of SJ and S | 266 |

7 Classes H and N | 273 |

Power series of bounded variation | 285 |

Cauchys integral | 288 |

Conformal mapping | 289 |

Miscellaneous theorems and examples | 295 |

CHAPTER VIII | 298 |

Further examples of divergent Fourier series | 302 |

Examples of Fourier series divergent almost everywhere | 305 |

4 An everywhere divergent Fourier series | 310 |

Miscellaneous theorems and examples | 314 |

CHAPTER IX | 316 |

2 Formal integration of series | 319 |

3 Uniqueness of the representation by trigonometric series | 325 |

The principle of localization Formal multiplication of trigonometric | 330 |

Formal multiplication of trigonometric series cont page | 337 |

8 Uniqueness of summable trigonometric series cont | 356 |

Notes | 375 |

4 Marcinkiewiczs theorem on the interpolation of operations page 11 | 111 |

5 Paleys theorems on Fourier coefficients 12 | 121 |

Theorems of Hardy and Littlewood about rearrangements of Fourier coefficients | 127 |

Lacunary coefficients | 132 |

Fractional integration | 137 |

9 Fractional integration cont | 138 |

FourierStieltjes coefficients 11 FourierStieltjes coefficients and sets of constant ratio of dissection 1 | 147 |

Miscellaneous theorems and examples 1 | 157 |

CHAPTER XIII | 161 |

Partial sums of Sf for | 165 |

Order of magnitude of Sn for e | 167 |

A test for the convergence of S almost everywhere | 170 |

Majorants for the partial sums of S and S 5 Behaviour of the partial sums of S and S 6 Theorems on the partial sums of power series | 179 |

Theorems on the convergence of orthogonal series | 189 |

11 Capacity of sets and convergence of Fourier series | 194 |

Miscellaneous theorems and examples | 197 |

CHAPTER XIV | 199 |

2 The function a0 | 215 |

4 Convergence of conjugate series 21 | 216 |

5 The Marcinkiewicz function fi6 21 | 219 |

Miscellaneous theorems and examples 22 | 221 |

### Common terms and phrases

absolute constant absolutely continuous absolutely convergent analogue apply argument belongs bounded function bounded variation Chapter completes the proof conjugate consider continuous function converges absolutely converges almost everywhere converges uniformly convex deduce defined denote denumerable derivative diverges equal example exceed finite interval finite number fixed formula Fourier coefficients Fourier series Fourier-Stieltjes function F(x Hence holds hypothesis implies infinite kernel lacunary lacunary series Lebesgue left-hand side LEMMA linear log+ means modulus of continuity monotone necessary and sufficient neighbourhood non-decreasing non-negative non-tangential limit observe obtain partial sums particular periodic Poisson integral polynomial positive measure positive numbers power series prove regular replaced respectively result right-hand side S[dF satisfies condition sequence set of points Similarly Sn(x sufficient condition summable summation Suppose trigonometric polynomial trigonometric series uniform convergence uniformly bounded vanishes write zero Zygmund