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Author(s): Elias M. Stein, Rami Shakarchi
Publisher: World Scientific
Year: 2011
Language: English
Pages: 442
Tags: Математика;Функциональный анализ;
5.1 Separation of convex sets......Page 35
5.2 The Hahn-Banach Theorem......Page 39
5.3 Some consequences......Page 40
5.4 The problem of measure......Page 42
6 Complex L^p and Banach spaces......Page 46
7 Appendix: The dual of C(X)......Page 47
7.1 The case of positive linear functionals......Page 48
7.2 The main result......Page 51
7.3 An extension......Page 52
8 Exercises......Page 53
9 Problems......Page 62
2 Spaces in Harmonic Analysis......Page 66
1 Early Motivations......Page 67
2 The Riesz interpolation theorem......Page 71
2.1 Some examples......Page 76
3.1 The L^2 formalism......Page 80
3.2 The L^p theorem......Page 83
3.3 Proof of Theorem 3.2......Page 85
4 The maximal function and weak-type estimates......Page 89
4.1 The L^p inequality......Page 90
5 The Hardy space......Page 92
5.1 Atomic decomposition of Hr......Page 93
5.2 An alternative definition of Hr......Page 100
5.3 Application to the Hilbert transform......Page 101
6 The space Hr and maximal functions......Page 103
6.1 The space BMO......Page 105
7 Exercises......Page 109
8 Problems......Page 113
3 Distributions: Generalized Functions......Page 117
1 Elementary properties......Page 118
1.1 Definitions......Page 119
1.2 Operations on distributions......Page 121
1.3 Supports of distributions......Page 123
1.4 Tempered distributions......Page 124
1.5 Fonrier transform......Page 126
1.6 Distributions with point supports......Page 129
2.1 The Hubert transform and pv(1/r)......Page 130
2.2 Homogeneous distributions......Page 134
2.3 Fundamental solutions......Page 144
2.4 Fundamental solution to general partial differential equations with constant coefficients......Page 148
2.5 Parametrices and regularity for elliptic equations......Page 150
3.1 Defining properties......Page 153
3.2 The L^p theory......Page 157
4 Exercises......Page 164
5 Problems......Page 172
4 Applications of the Baire Category Theorem......Page 176
1 The Baire category theorem......Page 177
1.1 Continuity of the limit of a sequence of continuous functions......Page 179
1.2 Continuous functions that are nowhere differentiable......Page 182
2 The uniform boundedness principle......Page 185
2.1 Divergence of Fourier series......Page 186
3 The open mapping theorem......Page 189
3.1 Decay of Fourier coefficients of L^1-functions......Page 192
4.1 Grothendieck's theorem on closed subspaces of L^p......Page 193
5 Besicovitch sets......Page 195
6 Exercises......Page 200
7 Problems......Page 204
5 Rudiments of Probability Theory......Page 207
1.1 Coin flips......Page 208
1.2 The case N = \infinity......Page 210
1.3 Behavior of as N —+ \infinity first results......Page 213
1.4 Central limit theorem......Page 214
1.5 Statement and proof of the theorem......Page 216
1 .6 Random series......Page 218
1.7 Random Fourier series......Page 221
1.8 Bernoulli trials......Page 223
2.1 Law of large numbers and ergodic theorem......Page 224
2.2 The role of martingales......Page 227
2.4 The central limit theorem......Page 234
2.5 Random variables with values in R^d......Page 239
2.6 Random walks......Page 241
3 Exercises......Page 246
4 Problems......Page 254
6 An Introduction to Brownian Motion......Page 257
1 The Framework......Page 258
2 Technical Preliminaries......Page 260
3 Construction of Brownian motion......Page 265
4 Some further properties of Brownian motion......Page 270
5 Stopping times and the strong Markov property......Page 272
5.1 Stopping times and the Blumenthal zero-one law......Page 273
5.2 The strong Markov property......Page 277
5.3 Other forms of the strong Markov Property......Page 279
6 Solution of the Dirichiet problem......Page 283
7 Exercises......Page 287
8 Problems......Page 292
1 Elementary properties......Page 295
2 Hartogs' phenomenon: an example......Page 299
3 Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations......Page 302
4 A boundary version: the tangential Cauchy-Riemann equations......Page 307
5 The Levi form......Page 312
6 A maximum principle......Page 315
7 Approximation and extension theorems......Page 318
8 Appendix: The upper half-space......Page 326
8.1 Hardy space......Page 327
8.2 Cauchy integral......Page 330
8.3 Non-solvability......Page 332
9 Exercises......Page 333
10 Problems......Page 338
8 Oscillatory Integrals in Fourier Analysis......Page 340
I An illustration......Page 341
2 Oscillatory integrals......Page 344
3 Fourier transform of surface-carried measures......Page 351
4 Return to the averaging operator......Page 356
5.1 Radial functions......Page 362
5.3 The theorem......Page 364
6.1 The Schrödinger equation......Page 367
6.2 Another dispersion equation......Page 371
6.3 The non-homogeneous Schrödinger equation......Page 374
6.4 A critical non-linear dispersion equation......Page 378
7.1 A variant of the Radon transform......Page 382
7.2 Rotational curvature......Page 384
7.3 Oscillatory integrals......Page 386
7.4 Dyadic decomposition......Page 389
7.5 Almost-orthogonal sums......Page 392
7.6 Proof of Theorem 7.1......Page 393
8 Counting lattice points......Page 395
8.1 Averages of arithmetic functions......Page 396
8.2 Poisson summation formula......Page 398
8.3 Hyperbolic measure......Page 403
8.4 Fourier transforms......Page 408
8.5 A summation formula......Page 411
9 Exercises......Page 417
10 Problems......Page 424
Notes and References......Page 428
Bibliography......Page 432
Symbol Glossary......Page 436
Index......Page 438