| 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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