Trigonometric Series, Volume I & II Combined

Cover......Page 1
Foreword by Fefferman......Page 4
Half Title......Page 12
Copyright......Page 15
Dedication......Page 16
Preface......Page 24
Lint of Symbols......Page 27
Title Page......Page 14
CONTENTS......Page 18
1.  Trigonometric series......Page 28
2. Summation by parts......Page 30
3. Orthogonal series......Page 32
4. The trigonometric system......Page 33
5. Fourier-Stieltjes series......Page 37
6. Completeness of the trigonometric system......Page 38
7. Bessel's inequality and Parseval's formula......Page 39
8. Remarks on series and integrals......Page 41
9. Inequalities......Page 43
10. Convex functions......Page 48
11. Convergence in Lr......Page 53
12. Sets of the first and second categories......Page 55
13. Rearrangements of functions. Maximal theorems of Hardy and Littlewood......Page 56
Miscellaneous theorems and examples......Page 61
1. Formal operations on S[ f J......Page 62
2. Differentiation and integration of S[f]......Page 67
3. Modulus of continuity. Smooth functions......Page 69
4. Order of magnitude of Fourier coefficients......Page 72
5. Formulae for partial sums of S[f ] and S[f ]......Page 76
6. The Dini test and the principle of localization......Page 79
7. Some more formulae for partial sums......Page 82
8. The Dirichlet-Jordan test......Page 84
9. Gibbs's phenomenon......Page 88
10. The Dini-Lipschitz test......Page 89
11. Lebesgue's test......Page 92
12. Lebesgue constants......Page 94
13. Poisson's summation formula......Page 95
Miscellaneous theorems and examples......Page 97
1. Summability of numerical series......Page 101
2. General remarks about the summability of S[f ] and f ]......Page 111
3. Summability of S[f ] and S[f ] by the method of the first arithmetic mean......Page 115
4. Convergence factors......Page 120
5. Summability (C, a)......Page 121
6. Abel summability......Page 123
7. Abel summability (cont.)......Page 126
8. Summability of S[dF] and S[dF]......Page 132
9. Fourier series at simple discontinuities......Page 133
10. Fourier sine series......Page 136
11. Gibbs's phenomenon for the method (C, a)......Page 137
12. Theorems of Rogosinski......Page 139
13. Approximation to functions by trigonometric polynomials......Page 141
Miscellaneous theorems and examples......Page 151
1. The class L^2......Page 154
2. A theorem of Marcinkiewicz......Page 156
3. Existence of the conjugate function......Page 158
4. Classes of functions and (C, 1) means of Fourier series......Page 163
5. Classes of functions and (C, I) means of Fourier series (cont.)......Page 170
6. Classes of functions and Abel means of Fourier series......Page 176
7. Majorants for the Abel and Cesaro means of S[f]......Page 181
8. Parseval's formula......Page 184
9. Linear operations......Page 189
10. Classes L^m......Page 197
11. Conversion factors for classes of Fourier series......Page 202
Miscellaneous theorems and examples......Page 206
1. Series with coefficients tending monotonically to zero......Page 209
2. The order of magnitude of functions represented by series with  monotone coefficients......Page 213
3. A class of Fourier-Stieltjes series......Page 221
4. The series En-i-a eicn lo[ n einx......Page 224
5. The series 2,v-1eil'e'......Page 227
6. Lacunary series......Page 229
7. Riesz products......Page 235
8. Rademacher series and their applications......Page 239
9. Series with `small' gaps......Page 249
10. A power series of Salem......Page 252
Miscellaneous theorems and examples......Page 255
1. General series......Page 259
2. Sets N......Page 263
3. The absolute convergence of Fourier series......Page 267
4. Inequalities for polynomials......Page 271
5. Theorems of Wiener and Levy......Page 273
6. The absolute convergence of lacunary series......Page 274
3. Applications of Green's formula......Page 287
1. Existence of conjugate functions page......Page 279
2. The Fourier character of conjugate series......Page 280
4. Integrability B......Page 289
5. Lipschitz conditions......Page 290
6. Mean convergence of S[f] and S[ f ]......Page 293
7. Classes HD and N......Page 298
8. Power series of bounded variation......Page 312
9. Cauchy's integral......Page 315
10. Conformal mapping......Page 316
Miscellaneous theorems and examples......Page 322
1. Divergence of Fourier series of continuous functions......Page 325
2. Further examples of divergent Fourier series......Page 329
3. Examples of Fourier series divergent almost everywhere......Page 332
4. An everywhere divergent Fourier series......Page 337
Miscellaneous theorems and examples......Page 341
1. General remarks. The Cantor-Lebesgue theorem......Page 343
2. Formal integration of series......Page 346
3. Uniqueness of the representation by trigonometric series......Page 352
4. The principle of localization. Formal multiplication of trigonometric series......Page 357
5. Formal multiplication of trigonometric series (cont.) page.......Page 364
6. Sets of uniqueness and sets of multiplicity......Page 371
7. Uniqueness of summable trigonometric series......Page 379
S. Uniqueness of summable trigonometric series (cont.)......Page 383
9. Localization for series with coefficients not tending to zero......Page 390
Miscellaneous theorems and examples......Page 397
Notes......Page 402
Title Page......Page 411
CONTENTS......Page 413
1. General remarks page......Page 417
2. Interpolating polynomials as Fourier series......Page 422
3. The case of an even number of fundamental points......Page 424
4. Fourier-Lagrange coefficients......Page 430
5. Convergence of interpolating polynomials......Page 432
6. Jackson polynomials and related topics......Page 437
7. Mean convergence of interpolating polynomials......Page 443
8. Divergence of interpolating polynomials......Page 451
9. Divergence of interpolating polynomials (cont.)......Page 460
10. Polynomials conjugate to interpolating polynomials......Page 464
Miscellaneous theorems and examples......Page 471
1. Ceei ro summability of differentiated series......Page 475
2. Summability C of Fourier series......Page 481
3. A theorem on differentiated series......Page 487
4. Theorems on generalized derivatives......Page 489
5. Applications of Theorem (4.2) to Fourier series......Page 496
6. The integral M and Fourier series......Page 499
7. The integral M=......Page 502
Miscellaneous theorems and examples......Page 507
1. The Riesz-Thorin theorem......Page 509
2. The theorems of Hausdorff-Young and F. Riesz......Page 517
3. Interpolation of operations in the olosees Hr......Page 521
4. Marcinkiewiez's theorem on the interpolation of operations page......Page 527
5. Paley's theorems on Fourier ooefcients......Page 536
6. Theorems of Hardy and Littlewood about rearrangements of Fourier coefficients......Page 543
7. Lacunary coefficients......Page 547
8. Fractional integration......Page 549
9. Fractional integration (cont.)......Page 554
10. Fourier-Stieltjes coefficients......Page 558
11. Fourier-Stieltjee coefficients and sets of oonetsnt ratio of dissection......Page 563
Miscellaneous theorems and examples......Page 572
1. Partial sums of S[ f ] for f e L'......Page 577
.2 Order of magnitude of S. for f e LP......Page 602
3. A test for the convergence of S f f ] almost everywhere......Page 586
4. Majorants for the partial sums of SL f] and 4f]......Page 589
5. Behaviour of the partial sums of Sf f ] and f f j......Page 591
6. Theorems on the partial sums of power series......Page 594
7. Strong summability of Fourier series. The case f e L/, r > I......Page 596
8. Strong summability of S[f) and f f) in the general case......Page 600
9. Almost convergence of S[f] and S[f]......Page 604
10. Theorems on the convergence of orthogonal series......Page 605
11. Capacity of sets and convergence of Fourier series......Page 610
Miscellaneous theorems and examples......Page 613
1. Boundary behaviour of harmonic and analytic functions......Page 615
2. The function s(6)......Page 623
3. The Littlewood-Paley function g(B)......Page 626
4. Convergence of conjugate series......Page 632
5. The Marcinkiewicz function p(8)......Page 635
Miscellaneous theorems and examples......Page 637
1. General remarks page......Page 638
2. Functions in If, 1 < r < m......Page 640
3. Functions in If, 1 < r < oo (cont.)......Page 645
4. Theorems on the partial sums of S[f ), f c L', 1 < r < oo......Page 646
5. The limiting case r = 1......Page 650
6. The limiting case r = co......Page 655
1. General remarks......Page 658
2. Fourier transforms......Page 662
3. Fourier transforms (cont.)......Page 670
4. Fourier-Stieltjee bransforms......Page 674
5. Applications to trigonometric series......Page 679
6. Applications to trigonometric series (cont.)......Page 685
7. The Paley-Wiener theorem......Page 688
8. Riemann theory of trigonometric integrals......Page 694
9. Equiconvergence theorems......Page 702
10. Problems of uniqueness......Page 707
 Miscellaneous theorems and examples......Page 713
1. General remarks......Page 716
2. Strong differentiability of multiple integrals and its applications......Page 721
3. Restricted summability of Fourier series......Page 725
4. Power series of several variables......Page 731
5. Power series of several variables (cont.)......Page 737
Miscellaneous theorems and examples......Page 744
Notes......Page 747
Bibliography......Page 752
Index......Page 769
Back Cover......Page 781