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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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47 289 15 54 289 4 19 235 306 308 11 266 6 11 32 11 173 11 11 424 182 16 320 35 53 95 18 83 5 2 5 5 5 5 10 10 8 182 182 13 6 385 21 278 10 10 (C,1) summable 228 304 A*, adjoint 135 a.e. 6 Abel, continuity theorem 232 Abel, kernel 283 288 318 Abel, summability 231 Absolutely, continuous 167 213 Absolutely, convergent 63 Absolutely, summable 63 Adherence point 52 Adjoint in Hilbert space 136 Adjoint, map A* 135 Admissibility condition 484 Algebraic complement 85 Almost everywhere 6 Angle in inner product space 15 Antipodal points 47 Approximate identity for 330 Approximate identity for 271 Approximation, best 90 Approximation, best mean 99 B(x,r), open ball 35 B-spline 476 Baire null space complete 66 Baire, category theorem 61 Baire, null space 8 Balian — Low theorem 413 Ball, closed 35 Ball, open 35 Banach — Steinhaus theorem 211 Banach, space 61 Band-limited 281 290 Base = basis, Hamel 21 Base = basis, Riesz 450 Base = basis, Schauder 450 Basis problem 89 Battle — Lemarie wavelets 478 Bessel, equality 20 23 103 Bessel, inequality 24 103 256 Best approximation 90 98 Best approximation from complete subspace 92 Best approximation, does not exist 94 97 Best mean approximation 101 Bicontinuous 39 Big Oh notation 223 Bijective 7 Bisequences 5 Biseries 5 Bit reversal 405 Bolzano — Weierstrass theorem 71 Bounded variation 164 Bounded, linear map 50 Bounded, map 49 Bounded, metric 81 Bounded, operator 126 Bounded, set 49 Buneman's algorithm 405 Butterfly relationship 403 BV[a,b] 164 C(x,r), closed ball 35 C, complex numbers 1 CA, complement of A 58 Cantor — Lebesgue theorem 172 Carleson 211 Carleson, pointwise convergence theorem 194 Cauchy — Schwarz 14 16 Cauchy — Schwarz, equality in 14 Cauchy, principal value 182 Cauchy, sequence 60 Causal 341 Cesaro kernel 282 318 Cesaro kernel, Fourier transform of 318 Cesaro summability 228 CHARACTER 396 Characteristic function 6 278 Chebyshev, polynomial 244 Chebyshev, subspace 90 Circle group T 265 Circulant 394 cl, closure operator 52 Clopen 59 Closed, map 75 Closed, set 52 Closure 52 Cluster point 45 Codimension 85 Coefficient functionals 456 Commutatively convergent 119 Compact 71 Compact, sequentially 71 Compact, support 54 290 436 complete 61 Complete orthonormal set 107 Complete sequence 465 Completeness and nested sequences 64 Completion 61 Conditionally convergent 119 Conjugate symmetry 13 Conjugate, series 197 Conjugate, space 131 Continuity 59 Continuity in the mean 73 260 Continuity of a function 39 Continuity of a linear map 43 Continuity of basic operations 40 Continuity of linear map 47 Continuity of projections 46 Continuity, joint 46 Continuous, dual 131 Continuous, Fourier kernel 177 Continuous, not differentiable 161 Continuous, wavelet transform (CWT) 483 Convergence in metric space 38 Convergence in the mean 43 Convergence of series 46 Convergence of two-sided series 46 Convergence on products 42 Convergence, commutative 119 Convergence, componentwise 41 Convergence, conditional 119 Convergence, equivalents 38 Convergence, mean 43 Convergence, pointwise 43 Convergence, unconditional 119 Convergence, uniform 44 Convex 17 Convolution 268 328 370 Countable 55 Coz x, cozero set of x 19 Cozero set 19 Cyclic 397 Cyclic convolution 397 Cyclic on 291 Cyclic, Fourier cosine transform 350 Cyclic, theorem 269 293 328 371 397 C[a,b] 11 d(x,S) 57 de la Valee-Poussin pointwise convergence theorem 195 Decapitation 227 Decomposition 432 Dense 52 Dense, subspaces of 54 Derivative one-sided 165 DFT 385 DFT, matrix 386 Diameter 57 64 Diameter (of a set) 51 Diameter, finite for compact space 74 Dido 224 Dilation 114 411 Dilation equation 416 Dimension of vector space 22 Dini Test 186 Dini Theorem 298 Direct product 84 Direct sum 123 Direct sum, external 83 124 Direct sum, inner product 83 Direct sum, internal 85 123 Direct sum, topological 86 Dirichlet, function 161 Dirichlet, integral 182—183 Dirichlet, kernel 174 178 Dirichlet, pointwise convergence theorem 191 Dirichlet, test for series of constants 217 Dirichlet, test for series of functions 218 Discrete Fourier Transform 385 391 Discrete Fourier transform, inversion 390 discrete wavelet transform (DWT) 484 Distance from a point to a set 57 Distance function 1 Dominated Convergence Theorem 44 172 Dual, complete 134 Dual, frame representation 468 Dual, second 134 Dual, space 131 Dvoretzky — Rogers theorem 120 e-net 69 Eigenvalue 137 Eigenvector 137 Enflo 90 Equivalent, metrics 81 Equivalent, norms 78 Equivalent, Schauder bases 450 Essential supremum 33 Essentially bounded 33 Euclidean norm 10 Euclidean space 2 Evaluation map 131 Even function 19 147 153 Even function, odd derivative 69 Even part of function 400 Extension by continuity 67 Exterior of set 59 External direct sum 124 Fast Fourier transform algorithm (FFT) 400 Fatou's lemma 439 Fejer kernel 235 Fejer kernel, continuous 308 Fejer kernel, n-dimensional 258 Fejer theorem 238 259 Fejer theorem, n-dimensional 259 FFT for 401 FFT for N = RC 408 FFT, algorithm 400 Filter 423—424 Filter and Fourier transform 424 Filter, compact support 437 Filter, Haar 424 Filter, low-pass 423—424 Filter, orthonormality of translates 442 Finite Fourier mapping 266 Finite Fourier transform 265 Finite FT 265 Finite sequences 51 Fourier coefficients 104 112 144 Fourier coefficients, absolute convergence 203 Fourier coefficients, products 153 Fourier coefficients, size 157 167 222 Fourier cosine integral 302 Fourier cosine series 155 Fourier cosine transform 341 Fourier cosine transform in 366 Fourier integral formula 276 295—297 Fourier kernel, continuous 111 Fourier kernel, discrete 179 Fourier kernel, selector property 190 193 368 Fourier map 289—290 Fourier map, finite 266 Fourier map, for 363 Fourier map, injective 313 Fourier map, not onto 314 Fourier series 144 Fourier series in Hilbert space 112 Fourier series of dilation 151 Fourier series, A-summability 240 249 Fourier series, amplitude-phase 151 Fourier series, complex form 144 Fourier series, convergence 211 Fourier series, divergent 211 Fourier series, exponential form 144 151 264 Fourier series, higher dimensions 249 Fourier series, period 2p 149 Fourier series, pointwise convergence 184 239 246 257 Fourier series, pointwise convergence for BV 194 Fourier series, several variables 253 Fourier series, speed of convergence 222 Fourier series, table of convergence 206 249 Fourier series, termwise differentiation 221 Fourier series, termwise integration 214 Fourier series, time-shift 152 Fourier series, uniform convergence 203 206 239 300 Fourier sine series 155 Fourier sine transform 341 Fourier transform 278 289 Fourier transform of derivative 334 Fourier transform of integral 336
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![$C_{1}[a,b]$](/math_tex/3ddd9297435451d452c86913d0b1711e82.gif)
, standard basis element 
, Fejer kernel 
, continuous Fejer kernel 
, only real in Ch. 4. 
![$L_{p}[a,b]$](/math_tex/3acd0831134354fa62ee92bde72ee80582.gif)
smaller for bigger p 
, real-valued 
, real-valued 
, filter associated with g 
, nonnegative real numbers 
, sphere 
, orthocomplement 
, direct sum 
inner product 
, indicator of rationals 
, Fourier transform 
![$L_{p}[-\pi,\pi],p\geq1$](/math_tex/ed748f1f1f987454367133c1e009ac3182.gif)
functions 
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