Representation Theorems in Hardy Spaces

by Javad Mashreghi

• ISBN13: 9780521732017
• ISBN10: 0521732018
• Edition: 1st
• Format: Paperback
• Copyright: 2009-04-20
• Publisher: Cambridge University Press

Summary

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, probability theory, and operator theory, and has found essential applications in robust control engineering.

Table of Contents

Preface	p. xi
Fourier series	p. 1
The Laplacian	p. 1
Some function spaces and sequence spaces	p. 5
Fourier coefficients	p. 8
Convolution on T	p. 13
Young's inequality	p. 16
Abel-Poisson means	p. 21
Abel-Poisson means of Fourier series	p. 21
Approximate identities on T	p. 25
Uniform convergence and pointwise convergence	p. 32
Weak* convergence of measures	p. 39
Convergence in norm	p. 43
Weak* convergence of bounded functions	p. 47
Parseval's identity	p. 49
Harmonic functions in the unit disc	p. 55
Series representation of harmonic functions	p. 55
Hardy spaces on D	p. 59
Poisson representation of h∞(D) functions	p. 60
Poisson representation of hp(D) functions (1 < p < ∞)	p. 65
Poisson representation of h1(D) functions	p. 66
Radial limits of hp(D) functions (1 ≤ p ≤ ∞)	p. 70
Series representation of the harmonic conjugate	p. 77
Logarithmic convexity	p. 81
Subharmonic functions	p. 81
The maximum principle	p. 84
A characterization of subharmonic functions	p. 88
Various means of subharmonic functions	p. 90
Radial subharmonic functions	p. 95
Hardy's convexity theorem	p. 97
A complete characterization of hp(D) spaces	p. 99
Analytic functions in the unit disc	p. 103
Representation of Hp(D) functions (1 < p ≤ ∞)	p. 103
The Hilbert transform on T	p. 106
Radial limits of the conjugate function	p. 110
The Hilbert transform of C1(T) functions	p. 113
Analytic measures on T	p. 116
Representations of H1(D) functions	p. 120
The uniqueness theorem and its applications	p. 123
Norm inequalities for the conjugate function	p. 131
Kolmogorov's theorems	p. 131
Harmonic conjugate of h2(D) functions	p. 135
M. Riesz's theorem	p. 136
The Hilbert transform of bounded functions	p. 142
The Hilbert transform of Dini continuous functions	p. 144
Zygmund's L log L theorem	p. 149
M. Riesz's L log L theorem	p. 153
Blaschke products and their applications	p. 155
Automorphisms of the open unit disc	p. 155
Blaschke products for the open unit disc	p. 158
Jensen's formula	p. 162
Riesz's decomposition theorem	p. 166
Representation of Hp(D) functions (0 < p < 1)	p. 168
The canonical factorization in Hp(D) (0 < p ≤ ∞)	p. 172
The Nevanlinna class	p. 175
The Hardy and Fejér-Riesz inequalities	p. 181
Interpolating linear operators	p. 187
Operators on Lebesgue spaces	p. 187
Hadamard's three-line theorem	p. 189
The Riesz-Thorin interpolation theorem	p. 191
The Hausdorff-Young theorem	p. 197
An interpolation theorem for Hardy spaces	p. 200
The Hardy-Littlewood inequality	p. 205
The Fourier transform	p. 207
Lebesgue spaces on the real line	p. 207
The Fourier transform on L1(R)	p. 209
The multiplication formula on L1(R)	p. 218
Convolution on R	p. 219
Young's inequality	p. 221
Poisson integrals	p. 225
An application of the multiplication formula on L1(R)	p. 225
The conjugate Poisson kernel	p. 227
Approximate identities on R	p. 229
Uniform convergence and pointwise convergence	p. 232
Weak* convergence of measures	p. 238
Convergence in norm	p. 241
Weak* convergence of bounded functions	p. 243
Harmonic functions in the upper half plane	p. 247
Hardy spaces on C+	p. 247
Poisson representation for semidiscs	p. 248
Poisson representation of h(&Cbar;+) functions	p. 250
Poisson representation of hp(C+) functions (1 ≤ p ≤ ∞)	p. 252
A correspondence between &Cbar;+ and &Dbar;	p. 253
Poisson representation of positive harmonic functions	p. 255
Vertical limits of hp(C+) functions (1 ≤ p ≤ ∞)	p. 258
The Plancherel transform	p. 263
The inversion formula	p. 263
The Fourier-Plancherel transform	p. 266
The multiplication formula on Lp(R) (1 ≤ p ≤ 2)	p. 271
The Fourier transform on Lp(R) (1 ≤ p ≤ 2)	p. 273
An application of the multiplication formula on Lp(R) (1 ≤ p ≤ 2)	p. 274
A complete characterization of hp(C+) spaces	p. 276
Analytic functions in the upper half plane	p. 279
Representation of Hp(C+) functions (1 < p ≤ ∞)	p. 279
Analytic measures on R	p. 284
Representation of H1(C+) functions	p. 286
Spectral analysis of Hp(R) (1 ≤ p ≤ 2)	p. 287
A contraction from Hp(C+) into Hp(D)	p. 289
Blaschke products for the upper half plane	p. 293
The canonical factorization in Hp(C+) (0 < p ≤ ∞)	p. 294
A correspondence between Hp(C+) and Hp(D)	p. 298
The Hilbert transform on R	p. 301
Various definitions of the Hilbert transform	p. 301
The Hilbert transform of C1c(R) functions	p. 303
Almost everywhere existence of the Hilbert transform	p. 305
Kolmogorov's theorem	p. 308
M. Riesz's theorem	p. 311
The Hilbert transform of Lip¿(t) functions	p. 321
Maximal functions	p. 329
The maximal Hilbert transform	p. 336
Topics from real analysis	p. 339
A very concise treatment of measure theory	p. 339
Riesz representation theorems	p. 344
Weak* convergence of measures	p. 345
C(T) is dense in Lp(T) (0 < p < ∞)	p. 346
The distribution function	p. 347
Minkowski's inequality	p. 348
Jensen's inequality	p. 349
A panoramic view of the representation theorems	p. 351
hp(D)	p. 352
h1(D)	p. 352
hp(D) (1 < p < ∞)	p. 354
h∞(D)	p. 355
Hp(D)	p. 356
Hp(D) (1 ≤ p < ∞)	p. 356
H∞(D)	p. 358
hp(C+)	p. 359
h1(C+)	p. 359
hp(C+) (1 < p ≤ 2)	p. 361
hp(C+) (2 < p < ∞)	p. 362
h∞(C+)	p. 363
h+(C+)	p. 363
Hp(C+)	p. 364
Hp(C+) (1 ≤ p ≤ 2)	p. 364
Hp(C+) (2 < p < ∞)	p. 365
H∞(C+)	p. 366
Bibliography	p. 367
Index	p. 369
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