Trigonometric Series, Volume 1Cambridge University Press, 2002 - 747 pages Professor 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 Volume II provides much material previously unpublished in book form. |
Table des matières
4 The trigonometric system | 5 |
4 Marcinkiewiczs theorem on the interpolation of operations page | 11 |
Paleys theorems on Fourier coefficients | 12 |
8 Remarks on series and integrals | 15 |
Convex functions | 21 |
Sets of the first and second categories | 28 |
Miscellaneous theorems and examples | 34 |
2 Differentiation and integration of Sf | 41 |
Miscellaneous theorems and examples | 228 |
CHAPTER VI | 232 |
Sets N | 235 |
3 The absolute convergence of Fourier series | 240 |
4 Inequalities for polynomials | 244 |
Theorems of Wiener and Lévy | 245 |
The absolute convergence of lacunary series | 247 |
Miscellaneous theorems and examples | 250 |
4 Order of magnitude of Fourier coefficients | 47 |
CHAPTER VII | 51 |
Formulae for partial sums of Sf and Šƒ | 55 |
9 Gibbss phenomenon page | 61 |
The DiniLipschitz test | 62 |
11 Lebesgues test | 65 |
12 Lebesgue constants | 67 |
13 Poissons summation formula | 68 |
Miscellaneous theorems and examples | 70 |
CHAPTER III | 74 |
General remarks about the summability of Sf and Šƒ | 84 |
3 Summability of Sƒ and ƒ by the method of the first arithmetic mean | 88 |
4 Convergence factors | 93 |
5 Summability C α | 94 |
Abel summability | 96 |
7 Abel summability cont | 99 |
8 Summability of SdF and SdF | 105 |
Fourier series at simple discontinuities | 106 |
Fourier sine series | 109 |
Gibbss phenomenon for the method C a | 111 |
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 |
Classes of functions and C 1 means of Fourier series cont | 143 |
6 Classes of functions and Abel means of Fourier series | 149 |
7 Majorants for the Abel and Cesaro means of Sf page | 155 |
8 Parsevals formula | 157 |
9 Linear operations | 162 |
Classes L | 170 |
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 Ena picn log n pinx | 197 |
The series Σvßeiva eivx | 200 |
6 Lacunary series | 202 |
7 Riesz products | 208 |
8 Rademacher series and their applications | 212 |
9 Series with small gaps | 222 |
A power series of Salem | 225 |
COMPLEX METHODS IN FOURIER SERIES 1 Existence of conjugate functions page | 252 |
2 The Fourier character of conjugate series | 253 |
3 Applications of Greens formula | 260 |
Integrability B | 262 |
Lipschitz conditions | 263 |
Mean convergence of Sƒ and Šƒ | 266 |
7 Classes HP and N | 271 |
Power series of bounded variation | 285 |
9 Cauchys integral | 288 |
Conformal mapping | 289 |
Miscellaneous theorems and examples | 295 |
CHAPTER VIII | 298 |
2 Further examples of divergent Fourier series | 302 |
3 Examples of Fourier series divergent almost everywhere | 305 |
4 An everywhere divergent Fourier series | 310 |
Miscellaneous theorems and examples | 314 |
CHAPTER IX | 316 |
Formal integration of series | 319 |
3 Uniqueness of the representation by trigonometric series | 325 |
4 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 |
Theorems of Hardy and Littlewood about rearrangements of Fourier | 6 |
coefficients 12 | 12 |
7 Lacunary coefficients 13 | 13 |
FourierStieltjes coefficients 14 | 14 |
Miscellaneous theorems and examples 15 | 15 |
CHAPTER XIII | 42 |
CONVERGENCE AND SUMMABILITY ALMOST EVERYWHERE | 161 |
1 Partial sums of Sf for fe L2 16 | 163 |
2 Order of magnitude of S for fe L 16 | 170 |
4 Majorants for the partial sums of Sƒ and ƒ 17 | 173 |
Behaviour of the partial sums of Sƒ and Šƒ 17 | 178 |
7 Strong summability of Fourier series The case fe L r1 18 | 180 |
8 Strong summability of Sƒ and ƒ in the general case 18 | 184 |
Theorems on the convergence of orthogonal series 18 | 189 |
Capacity of sets and convergence of Fourier series 19 | 194 |
Miscellaneous theorems and examples 19 | 197 |
CHAPTER XIV | 199 |
2 The function 80 | 210 |
3 The LittlewoodPaley function g0 21 | 219 |
Miscellaneous theorems and examples 22 | 221 |
Autres éditions - Tout afficher
Expressions et termes fréquents
a₁ absolutely continuous absolutely convergent apply arbitrarily argument b₁ belongs bounded variation Chapter completes the proof conjugate consider continuous function converges absolutely converges almost everywhere converges uniformly convex defined denote denumerable diverges einx eivx example exceed exists F is continuous F(eix f(x+t finite number follows formula Fourier coefficients Fourier series Fourier-Stieltjes function f(x Hence Hölder's inequality holds hypothesis implies infinite interval kernel lacunary series Lebesgue left-hand side LEMMA Let f(x lim sup limit linear means monotone n₁ necessary and sufficient non-decreasing non-negative observe obtain partial sums particular periodic Poisson integral polynomial positive measure positive numbers prove replaced result right-hand side S[dF satisfies sequence set of points Similarly Sn(x sufficient condition summation Suppose tends termwise THEOREM trigonometric series uniform convergence uniformly bounded zero Zygmund