Trigonometric Series, Volume 1

Front Cover
Cambridge University Press, 2002 - Mathematics - 747 pages
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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 Volume II provides much material previously unpublished in book form.
  

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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
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