Pinsky M.A. — Introduction to Fourier Analysis and Wavelets :: Электронная библиотека попечительского совета мехмата МГУ

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Pinsky M.A. — Introduction to Fourier Analysis and Wavelets

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Название: Introduction to Fourier Analysis and Wavelets

Автор: Pinsky M.A.

Аннотация:

This text for upper undergraduate or beginning graduate students provides an introduction to classical Fourier analysis and wavelets. A sampling of topics includes Fourier analysis on the circle, the n-dimensional Fourier transform, harmonic analysis, the Poisson summation formula, and probability theory. The volume contains more than 175 exercises, which are an integral part of the text. Pinsky developed the text while teaching at Northwestern University

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 376

Добавлена в каталог: 18.04.2008

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Предметный указатель

Abel means      45 64 261 266
Abel sums      49
Abelian theorem      54
Abel’s lemma      167
Absolutely continuous functions      21
Absolutely convergent trigonometric series      3 43
Almost-everywhere convergence      20 28 49 153 346
Analytic function      22 172
Approximate identitites      45 97
Averaging      118
Banach space      75
Banuelos, R      202
Beckner, W.      174 175
Benedicks, M.r      240
Bernoulli, D.      1
Bernstein’s inequality      122
Bernstein’s Theorem      43 70 129 350
Berry — Essen theorem      276
Bessel functions      6 81 140 154 164
Bessel’s inequality      36
Boas, R. P.      228
Bochner — Riesz summability      152
Bochner, S.      139
Bochner’s method of subordination      106
Bochner’s theorem on positive-definite functions      220 270
Borel — Cantelli Lemmas      281
Bounded linear operator      75
Bounded variation      27 60
Brando'ini, L.      149
Brownian motion      299
Calderon — Zygmund decomposition      209
Calderon — Zygmund theorem      211
Calderon, A. P.      194 196 209
Cantor measure      20
Cantor — Lebesgue theorem      87
Cantor’s uniqueness theorem      86
Carleson, L.      73 118
Central limit theorem      260
Cesaro average      119
Cohen’s theorem      338 262
Colzani, L.      149
Compactly supported wavelets      326
Completeness of the Haar functions      295
Completeness of the Hermite functions      138
Conjugate function      176
Conjugate Poisson kernel      8 179
Continuity theorem      269
Continuum wavelet      286
Contraction operator      14
Convergence of wavelet expansions      341
Convex function      170
Convex sequence      96
Convolution      5 14 19 92
Convolution semigroups      272
Daubechies’ formula      331
De la Vallee Poussin means      68
de — Moivre Laplace theorem      100
Dilation matrix      354
Dini condition      27 89
Dini’s theorem      115
Dirac measure      20
Directed set      46
Dirichlet kernel      15 25 1 12 140 223
Dirichlet — Jordan theorem      27 117
Dirichlet, P. G.      27
Distribution function      181
Divergence of Fourier series      73
du Bois — Reymond      73 1 17
Dyadic projections      292
Equidistribution mod      1 57
Euler formula      4
Exponential function      6
Factorial function      6
Fatou’s theorem      50 214
Fefferman, C.      73
Fejer means      45 54 62 98 120
Fejer — Riesz lemma      330
Finite measures      19 119 256 273
Fourier coefficient      4 13 19
Fourier cosine transform      124
Fourier reciprocity formula      4 14 93 97 231
Fourier sine transform      125
Fourier transforms      89 102 256 323
Fourier, J. B.      2
Gap series      262
Gauss — Weierstrass kernel      97
Gaussian approximation      80
Gaussian density function      95 257
Gaussian summability      104 232
Gaussian wavelet      288
Generalized h-transform      125
Gibbs — Wilbraham phenomenon      31 115
Grunbaum, A.      133
Haar function      387 306
Haar series      291 296
Haar wavelet      291 321
Hardy — Littlewood maximal function      197
Hardy, G. H.      59
Hardy’s Tauberian theorem      59
Harmonic functions      212
Hausdorff — Young inequality      174
Heat flow      2
Heat kernel      97 203 223
Hemandez — Weiss formula      333
Herglotz theorem      215
Hermite polynomials      134
Higher-order approximation in $C(\mathbb{T})$       66
Hilbert transform on $\mathbb{R}$       184 224
Hilbert transform on $\mathbb{T}$       176 224
Hoelder means      53
Hoelder’s inequality      170
Holder condition      13 40 63 116
Homogeneous Banach space      47 98
Hormander condition      211
Hunt, R.      73 118
Huygens’ principle      148
Independent random variables      261
Infinitely differentiable function      21
Integration of Fourier series      29
Isoperimetric inequality      38
Jackson’s theorem      65 348
Jordan, C.      27
Kac      266
Kahane, J. P.      45 73 143

Katznelson, I.      73 180
Kendall’s Theorem      241
Kolmogorov, A. N.      73 118 179
Kolmogorov’s inequality      192
Landau’s asymptotic formula      243
Laplace asymptotic method      6 9 80
Laplace’s equation      9 106 212
Large scale analysis      345
Lattice points      241
Laurent series      5
Law of the iterated logarithm      280
Lebesgue constants      61 75
Lebesgue Differentiation Theorem      200
Lebesgue set      124
Levy modulus of continuity      302
Levy — Khintchine Theorem      273
Lipschitz continuity      62
Lusin, N.      73
Marcinkiewicz interpolation theorem      206
Marcinkiewicz, J.      169 179
Maxwell’s theorem      109 259
Mexican hat wavelet      288
Meyer wavelets      318
Modified Bessel function      6 81 85
Modulus of continuity      12 20
Moore, C.      202
Multi-resolution analysis      303
Multiple Fourier series      230 244
Multiple harmonic motion      1
Negative binomial series      52
Newtonian potential kernel      109
Non-tangential convergence      202
Null-integrability      325
One-sided Fourier representation      124
Orthogonality      2 137
Orthonormal wavelet      319
Parseval’s identity      35
Periodic function      13
Periodization      223
Piecewise smooth function      21 145
Pinsky      139 145 147
Plancherel’s theorem      128
Pointwise convergence criteria      25 115 139 233 298 347
Poisson kernel      7 15 46 90 106 189
Poisson summation formula on      238
Poisson summation formula on R      1 225
Positive-definite function      257
Quotient groups      355
Radial functions      92 157 202 245
Random walk      252
Rapidly decreasing function      96
Rates of convergence in $C(\mathbb{T})$       61
Rates of convergence in $L^{2}(\mathbb{T})$       39
Restriction theorem      158
Riemann localization      31
Riemann means      64
Riemann — Lebesgue lemma      18 94
Riesz      73
Riesz kernels      195
Riesz potential kernel      107
Riesz systems      304 356
Riesz — Fischer theorem      37
Riesz — Thorin interpolation theorem      169
Rotations      194
Salem, R.      266
Scaling equation      310 357
Scaling filter      312
Scaling functions      304 319
Schrodinger equation and Gauss sums      247
Schwartz class      96
Schwartz distribution      146
Self-adjoint operator      15
Shannon sampling theorem      228
Shannon scaling function      306
simple harmonic motion      1
Simultaneous non-localization      240
Sine integral      24 114 185
Singular integrals      183 194 210
Skew-adjoint operator      16 176
Smith, К. T      202
Smoothness of wavelets      334
Sobolev Norm      43 290
Spherical Fourier inversion      139
Spherical maximal function      203
Spherical mean value      140
Spherical partial sum      139 233
Stationary phase method      162
Stein, E.      157 159 175 202 204 211
Stein’s complex interpolation method      159 175
Stirling’s formula      6
Strichartz, R.      33
Stronger summability method      52
Structure constants      310 357
Subgroups      354
Sublinear operator      197 207
Summability matrix      51
Summation by parts      11 79
Symmetric partial sum      4 112
Systems of wavelets      352
Tail sum      12
Tauberian theorem      54
Taylor, M. E      139 147 248
Tempered distribution      108 129
Three-lines theorem      172
Tomas, P.      159
Translation error      40
Uncertainty principle      134
Uniform boundedness principle      76 84
Uniform continuity      98
Uniform convergence      4
Uniqueness of Fourier coefficients      17
Upper half plane      216
Vibrating string      1
Wave equation      171
Wavelet transform      285
Wavelets      284
Weak convergence of measures      268 327
Weierstrass Af-test      4
Weiss, G      157
Whittaker, E. T      228
Wiener covering lemma      198
Wiener’s theorem      57 110
Young’s inequality for convolutions      174
Zygmund, A.      20 44 72 209 236 262 266
Zygmund’s theorem      44 209

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