Representation Theorems in Hardy Spaces

By: Javad Mashreghi

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Paperback | 18 May 2009

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

For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces : Hardy spaces of the open unit disc and Hardy spaces of the upper half plane.

With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

About the Author

Professor Javad Mashreghi is Bonyan Research Chair in Mathematical Analysis in the Department of Mathematics and Statistics at Laval University, Quebec. He won the prestigious G. de B. Robinson Award of the Canadian Mathematical Society in 2004 for two long research papers published in the Canadian Journal of Mathematics. His research interests are complex and harmonic analysis and their applications in applied sciences.

Industry Reviews

"Mathematicians working on related topics should find it a useful reference for statements and proofs of many of the classical results related the the Hardy spaces. Anyone teaching a course that includes Hardy spaces would find it a good source for homework problems." Peter Rosenthal, CMS Notes "... self-contained and clearly written text... The main strength of this book is a large number of exercises (over 300), which makes it a good textbook choice." Marcin M. Bownik, Mathematical Reviews

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Non-Fiction Mathematics Calculus & Mathematical Analysis Complex Analysis

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Non-Fiction Mathematics Calculus & Mathematical Analysis Complex Analysis

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 h^p(D) functions (1 < p < ∞)	p. 65
Poisson representation of h¹(D) functions	p. 66
Radial limits of h^p(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 h^p(D) spaces	p. 99
Analytic functions in the unit disc	p. 103
Representation of H^p(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 H^p(D) functions (0 < p < 1)	p. 168
The canonical factorization in H^p(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 h^p(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 h^p(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 L^p(R) (1 ≤ p ≤ 2)	p. 271
The Fourier transform on L^p(R) (1 ≤ p ≤ 2)	p. 273
An application of the multiplication formula on L^p(R) (1 ≤ p ≤ 2)	p. 274
A complete characterization of h^p(C₊) spaces	p. 276
Analytic functions in the upper half plane	p. 279
Representation of H^p(C₊) functions (1 < p ≤ ∞)	p. 279
Analytic measures on R	p. 284
Representation of H¹(C₊) functions	p. 286
Spectral analysis of H^p(R) (1 ≤ p ≤ 2)	p. 287
A contraction from H^p(C₊) into H^p(D)	p. 289
Blaschke products for the upper half plane	p. 293
The canonical factorization in H^p(C₊) (0 < p ≤ ∞)	p. 294
A correspondence between H^p(C₊) and H^p(D)	p. 298
The Hilbert transform on R	p. 301
Various definitions of the Hilbert transform	p. 301
The Hilbert transform of C¹_c(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 L^p(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
h^p(D)	p. 352
h¹(D)	p. 352
h^p(D) (1 < p < ∞)	p. 354
h^∞(D)	p. 355
H^p(D)	p. 356
H^p(D) (1 ≤ p < ∞)	p. 356
H^∞(D)	p. 358
h^p(C₊)	p. 359
h¹(C₊)	p. 359
h^p(C₊) (1 < p ≤ 2)	p. 361
h^p(C₊) (2 < p < ∞)	p. 362
h^∞(C₊)	p. 363
h⁺(C₊)	p. 363
H^p(C₊)	p. 364
H^p(C₊) (1 ≤ p ≤ 2)	p. 364
H^p(C₊) (2 < p < ∞)	p. 365
H^∞(C₊)	p. 366
Bibliography	p. 367
Index	p. 369
Table of Contents provided by Ingram. All Rights Reserved.

Representation Theorems in Hardy Spaces

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