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Bachman G., Beckenstein E. — Fourier And Wavelet Analysis
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Название: Fourier And Wavelet AnalysisАвторы: Bachman G., Beckenstein E. Аннотация: This book is intended as an introduction to classical Fourier analysis, Fourier series, and the Fourier transform. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2000Количество страниц: 506Добавлена в каталог: 30.11.2009Операции: Положить на полку |
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Предметный указатель
Fourier transform, conjugates 340 Fourier transform, discrete 385 Fourier transform, finite 144 265 Fourier transform, in 357 Fourier transform, integrability of 323 Fourier transform, inversion for BV 294 Fourier transform, windowed 480 Frame 465 Frame and Riesz basis 471 Frame, bounds 465 Frame, dual 468 Frame, exact 465 Frame, inequality 465 Frame, operator 465 467 Frame, tight 465 Frame, Weyl — Heisenberg 473 Franklin mother wavelet 462 Franklin wavelets 459 Frequently 45 Fubini theorem 251 Gab or transform 480 Gamma function 283 Gauss and FFT 399 Gaussian kernel 318 Gibbs phenomenon 208 Gibbs phenomenon, suppressed 242 Gram — Schmidt 21 Gregory series 192 Haar functions 115 Haar functions, density of in 119 Haar functions, density of in C[0,1] 118 Haar wavelets 420 Haar wavelets, filter 424 Haar wavelets, wavelets 412 Hahn — Banach theorem 131 Half-range expansion 155 Hardy — Landau theorem 240 Harmonics 145 Hat function 307 Hat function, Fourier transform 282 337 Hilbert cube 60 Hilbert cube, compact 74 Hilbert space 5 61 Hilbert space, dual of 132 Hilbert sum 124 Hoelder inequality 29 33 Homeomorphism 39 80 Homeomorphism, linear 80 Hunt 194 Idempotent 98 Identity none for 271 Indicator function 6 114 Inner product 13 Inner product isomorphic 25 Integral transform 263 Integration by parts 168 Interior of set 58 Interior point 58 Internal direct sum 123 Inversion theorem for 362 368 Inversion theorem for 312 Inversion theorem for BV functions 294 Inversion theorem for DFT 390 Inversion theorem, dominated 320 Inversion theorem, Fejer — Lebesgue 310 316 370 Inversion theorem, Fejer — Lebesgue for 370 Inversion theorem, Fourier cosine transform 350 Inversion theorem, Jordan 299 Inversion theorem, trigonometric form 298 Invertible element 398 Isometric 3 Isometry 3 Isoperimetric inequality 225 Joint continuity of metric and inner product 46 Jordan, decomposition theorem 165 Jordan, pointwise convergence theorem 194 240 275 299 K=R or C 2 Kernel 127 263 L(X,Y) 126 l.i.m., limit in the mean 43 Lebesgue point 246 Lebesgue pointwise convergence theorem 246 Leibniz's rule 335 Limit in the mean 43 Linear homeomorphism 80 Linear isometry 25 Linear isometry of 27 Linear isometry of C[a,b] 27 Linear isomorphism 24 Linear span 19 Linearly independent 20 Lipschitz condition 166 186 Lipschitz condition, uniform 166 Lipschitz test 186 299 Little oh notation 222 Localization principle 185 Low-pass filter 432 Mean approximation 99 Mean value theorem, Bonnet 193 Mean value theorem, first 196 Mean Value Theorem, second 193 196 Mean value theorem, second = Bonnet 193 Mellin transform 375 Menshov 249 Metric 1 Metric, bounded 81 Metric, equivalent 81 Metric, Euclidean 2 Metric, Hamming 8 Metric, max = sup 8 Metric, product 3 Metric, space 1 Metric, taxicab 4 Metric, trivial 5 Metric, uniform continuity of 68 Minkowski inequality 30 33 Monotone Convergence Theorem 324 Mother wavelet 411 419 Mother wavelet for Haar system 115 Mother wavelet for MRA 428 MRA 414 Multiresolution analysis (MRA) 414 Multiresolution analysis (MRA), Riesz 451 Neighborhood of a point 35 Neighborhood, basic 35 Nondifferentiable function 161 Norm 9 Norm isomorphic 25 Norm of 129 Norm of 129 Norm of a linear map 126 Norm of a projection 127 Norm of integral operator 127 Norm preserving 25 Norm, equivalent 78 Norm, max 12 Norm, reconstruct from unit ball 37 Norm, stronger 76 Norm, uniform 11 Norm, weaker 76 Normalized Lebesgue measure 264 Normed space 9 Nowhere dense 59 Null sequence 56 Null space 53 Nyquist, interval 374 Nyquist, rate 374 O, big oh 223 o, smaller order 222 Odd function 19 147 Odd part of function 399—400 Ondelette 419 Open set 58 Operators 126 Orthocomplement 53 95 Orthocomplement in Hilbert Space 96 Orthocomplement, properties 95 99 Orthogonal 18 Orthogonal, complement 53 95 Orthogonal, dimension 107 Orthogonal, direct sum 123 Orthogonal, projection 98 99 Orthogonal, set 18 Orthonormal basis 107 Orthonormal basis for 113 Orthonormal basis for 391 Orthonormal basis for 252 Orthonormal basis for 252 Orthonormal basis for 413 Orthonormal basis for 113 Orthonormal basis for 113 Orthonormal basis for 111 Orthonormal basis for band-limited 374 Orthonormal basis from scaling function 416 Orthonormal basis, pasting together 125 Orthonormal wavelet basis 419 Orthonormal, sequence 18 Orthonormal, set 18 Orthonormality of translates 422 432 p-Norm 10 p-norm, equivalent in finite-dimensional case 78 p-norm, inequivalent 79 P-summable 227 Parallelogram law 22 Parseval identity and energy 152 Parseval identity for 329 Parseval identity for 352 Parseval identity for 266 Parseval identity for 358 Parseval identity for 393 Parseval identity in Hilbert space 103—104 106 108 Parseval identity on dense subset 111 Parseval's equality 103 Periodic 143 Periodic extension odd 155 Periodic function integral 156 Periodic, doubly 250 Periodic, extension 143 250 Periodic, fundamental 153 Periodogram 394 Pickwickian 226 Piecewise, continuous 163 Piecewise, smooth 163 Plancherel, theorem 362 Plancherel, transform 357 Pointwise convergence theorem, de la Valee-Poussin 195 Pointwise convergence theorem, Dirichlet 191 Pointwise convergence theorem, Fejer 239 Pointwise convergence theorem, Jordan 194 240 275 Pointwise convergence theorem, Lebesgue 246 274 Pointwise convergence theorem, multiple 257 Poisson kernel 241 Polarization identity 23 Positive operator 466 Precompact 70 Principle of uniform boundedness 211 Product metrics 3 Projection map 46 Projection theorem 96 Projection, onto subspace 85 Projection, operator 42 Projection, orthogonal 98 Projection, uniform continuity of 69 Pseudometric 7 pth power summable 11 Pythagorean Theorem 20 23 105 Q, rationals 6 R, real numbers 1 Rademacher functions 109 Rapidly decreasing 363 Rare set 59 Rearrangement of series 119 Reciprocity theorem 361 Reconstruction 432 Reflexive space 134 Riemann — Lebesgue lemma 104 Riemann — Lebesgue lemma for 173 Riemann — Lebesgue lemma for 106 Riemann — Lebesgue lemma for 170 Riemann — Lebesgue lemma, variations 173 Riemann — Lebesgue property, Dirichlet kernel 179 Riemann — Lebesgue property, Fourier kernel 181 Riesz basis 450 Riesz basis and orthonormal basis 455 Riesz basis, multiresolution analysis 451 Riesz basis, scaling function 452 Riesz basis, translates of hat function 463 Riesz representation theorem 132 Riesz — Fischer theorem 104 Right angle in inner product space 16 Right-handed derivative 163 S(R) 363 s, summable sequences 27 Sampling theorem 373 Saw-tooth 149 Scalar homogeneous 133 Scaling function 415 Scaling function, compact support 445 Scaling function, Riesz 452 Scaling identity 115 425 Schauder base = basis 89 450 Schauder base = basis, equivalent 450 Sectionally continuous 163 Self-adjoint 135 Separable Hilbert space, 112 Separable Hilbert space, 121 Separable, iff complete orthonormal sequence 112 Separable, space 55 sgn t, signum function 109 Shannon wavelets 448 Shift matrix 395 Short-Time Fourier Transform 481 Signum function 109 Simple function 54 Sine integral 209 Smaller order 222 Solving linear equations 100 Spanned 19 Spectral 481 Sphere 35 SPLINE 476 Spline, basic 476—477 Square wave 147—148 Square-integrable functions 6 Square-summable functions 6 Square-summable sequences 5 Step function 54 Strict norm 15 Subspaces sums of 101 Summability kernel 318 Summability kernel and norm convergence 326 Summability kernel for 371 supp f 436 Support 435 T, circle group 265 Tauberian theorems 234 Taxicab metric 4 Time-limited 281 290 Toeplitz matrix 229 Toeplitz summable 230
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![$L_{2}[0,1]$](/math_tex/0ddf3918d7f9c7a4367f0c3e54e7736e82.gif)
functions 
![$C[-\pi,\pi]$](/math_tex/0f1785d555d887070d4bc9405c0101d282.gif)
![$L_{2}([-\pi,\pi]^{n})$](/math_tex/e1770d93b876aca9f9ca45628349f82e82.gif)
![$L_{2}([a,b]\times[c,d])$](/math_tex/d6ca1afd0188e9d942fb5a4a7778573082.gif)
![$L_{2}[0,\pi]$](/math_tex/c00bb0082d1567b7bef53955892daa5382.gif)
and ![$L_{2}[-\pi,\pi]$](/math_tex/a2525313dd7b70951e34648faf936bef82.gif)
![$f\in L_{1}[-\pi,\pi]$](/math_tex/e0ca231cb51a1768516eee81b6874fa282.gif)
![$f\in L_{1}[a,b]$](/math_tex/347b32d61c4b45aa2264c2a43379ec8782.gif)
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