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Representation Theorems in Hardy Spaces : London Mathematical Society Student Texts - Javad Mashreghi

Representation Theorems in Hardy Spaces

London Mathematical Society Student Texts

Paperback Published: 18th May 2009
ISBN: 9780521732017
Number Of Pages: 384

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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

Prefacep. xi
Fourier seriesp. 1
The Laplacianp. 1
Some function spaces and sequence spacesp. 5
Fourier coefficientsp. 8
Convolution on Tp. 13
Young's inequalityp. 16
Abel-Poisson meansp. 21
Abel-Poisson means of Fourier seriesp. 21
Approximate identities on Tp. 25
Uniform convergence and pointwise convergencep. 32
Weak* convergence of measuresp. 39
Convergence in normp. 43
Weak* convergence of bounded functionsp. 47
Parseval's identityp. 49
Harmonic functions in the unit discp. 55
Series representation of harmonic functionsp. 55
Hardy spaces on Dp. 59
Poisson representation of h(D) functionsp. 60
Poisson representation of hp(D) functions (1 < p < ∞)p. 65
Poisson representation of h1(D) functionsp. 66
Radial limits of hp(D) functions (1 ≤ p ≤ ∞)p. 70
Series representation of the harmonic conjugatep. 77
Logarithmic convexityp. 81
Subharmonic functionsp. 81
The maximum principlep. 84
A characterization of subharmonic functionsp. 88
Various means of subharmonic functionsp. 90
Radial subharmonic functionsp. 95
Hardy's convexity theoremp. 97
A complete characterization of hp(D) spacesp. 99
Analytic functions in the unit discp. 103
Representation of Hp(D) functions (1 < p ≤ ∞)p. 103
The Hilbert transform on Tp. 106
Radial limits of the conjugate functionp. 110
The Hilbert transform of C1(T) functionsp. 113
Analytic measures on Tp. 116
Representations of H1(D) functionsp. 120
The uniqueness theorem and its applicationsp. 123
Norm inequalities for the conjugate functionp. 131
Kolmogorov's theoremsp. 131
Harmonic conjugate of h2(D) functionsp. 135
M. Riesz's theoremp. 136
The Hilbert transform of bounded functionsp. 142
The Hilbert transform of Dini continuous functionsp. 144
Zygmund's L log L theoremp. 149
M. Riesz's L log L theoremp. 153
Blaschke products and their applicationsp. 155
Automorphisms of the open unit discp. 155
Blaschke products for the open unit discp. 158
Jensen's formulap. 162
Riesz's decomposition theoremp. 166
Representation of Hp(D) functions (0 < p < 1)p. 168
The canonical factorization in Hp(D) (0 < p ≤ ∞)p. 172
The Nevanlinna classp. 175
The Hardy and Fejér-Riesz inequalitiesp. 181
Interpolating linear operatorsp. 187
Operators on Lebesgue spacesp. 187
Hadamard's three-line theoremp. 189
The Riesz-Thorin interpolation theoremp. 191
The Hausdorff-Young theoremp. 197
An interpolation theorem for Hardy spacesp. 200
The Hardy-Littlewood inequalityp. 205
The Fourier transformp. 207
Lebesgue spaces on the real linep. 207
The Fourier transform on L1(R)p. 209
The multiplication formula on L1(R)p. 218
Convolution on Rp. 219
Young's inequalityp. 221
Poisson integralsp. 225
An application of the multiplication formula on L1(R)p. 225
The conjugate Poisson kernelp. 227
Approximate identities on Rp. 229
Uniform convergence and pointwise convergencep. 232
Weak* convergence of measuresp. 238
Convergence in normp. 241
Weak* convergence of bounded functionsp. 243
Harmonic functions in the upper half planep. 247
Hardy spaces on C+p. 247
Poisson representation for semidiscsp. 248
Poisson representation of h(&Cbar;+) functionsp. 250
Poisson representation of hp(C+) functions (1 ≤ p ≤ ∞)p. 252
A correspondence between &Cbar;+ and &Dbar;p. 253
Poisson representation of positive harmonic functionsp. 255
Vertical limits of hp(C+) functions (1 ≤ p ≤ ∞)p. 258
The Plancherel transformp. 263
The inversion formulap. 263
The Fourier-Plancherel transformp. 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+) spacesp. 276
Analytic functions in the upper half planep. 279
Representation of Hp(C+) functions (1 < p ≤ ∞)p. 279
Analytic measures on Rp. 284
Representation of H1(C+) functionsp. 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 planep. 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 Rp. 301
Various definitions of the Hilbert transformp. 301
The Hilbert transform of C1c(R) functionsp. 303
Almost everywhere existence of the Hilbert transformp. 305
Kolmogorov's theoremp. 308
M. Riesz's theoremp. 311
The Hilbert transform of Lip(t) functionsp. 321
Maximal functionsp. 329
The maximal Hilbert transformp. 336
Topics from real analysisp. 339
A very concise treatment of measure theoryp. 339
Riesz representation theoremsp. 344
Weak* convergence of measuresp. 345
C(T) is dense in Lp(T) (0 < p < ∞)p. 346
The distribution functionp. 347
Minkowski's inequalityp. 348
Jensen's inequalityp. 349
A panoramic view of the representation theoremsp. 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
Bibliographyp. 367
Indexp. 369
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521732017
ISBN-10: 0521732018
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 384
Published: 18th May 2009
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 14.986  x 2.032
Weight (kg): 0.522

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