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Bachman G., Beckenstein E. — Fourier And Wavelet Analysis
Bachman G., Beckenstein E. — Fourier And Wavelet Analysis



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Название: Fourier And Wavelet Analysis

Авторы: 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: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 506

Добавлена в каталог: 30.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$c_{0}$      47
$C_{0}(\mathbf{R})$      289
$C_{1}[a,b]$      15
$C_{c}(\mathbf{R})$      54 289
$D_{\infty}$      4
$e_{i}$, standard basis element      19
$F_{n}(t)$, Fejer kernel      235
$K_{r}(t)$, continuous Fejer kernel      306 308
$L_{1}(\mathbf{R})$      11
$L_{1}(\mathbf{T})$      266
$L_{2}(,R)$, only real in Ch. 4.      6
$L_{p}(E)$      11
$L_{p}(T,\mu)$      32
$L_{p}(\mathbf{R})$      11
$L_{p}[a,b]$ smaller for bigger p      173
$L_{p}^{r}(E)$, real-valued      11
$L_{p}^{r}(\mathbf{R})$, real-valued      11
$m_{g}$, filter associated with g      424
$PV\int^{b}_{a}$      182
$RH(S^{1})$      16
$R^{+}$, nonnegative real numbers      320
$S(x,r)$, sphere      35
$S^{\bot}$, orthocomplement      53 95
$x\bot y$      18
$X\oplus Y$, direct sum      83
$\ell_{2}$      5
$\ell_{2}(n)$      2
$\ell_{2}(\infty)$      5
$\ell_{2}(\mathbf{N})$      5
$\ell_{2}(\mathbf{Z})$      5
$\ell_{2}=\ell_{2}(\infty)=\ell_{2}(\mathbf{N})$      5
$\ell_{p}(n)$      10
$\ell_{p}(\mathbf{N})$      10
$\ell_{\infty}$      8
$\int^{\rightarrow b}_{a}$      182
$\int^{\rightarrow b}_{\rightarrow a}$      182
$\langle\ ,\ \rangle$ inner product      13
$\mathbf{1_{Q}}$, indicator of rationals      6
$\mathbf{Z}_{N}=\mathbf{Z}/(N)$      385
$\varphi$      21
$\widehat{f}(\omega)$, Fourier transform      278
$\|\ \|_{p}$      10
$\|\ \|_{\infty}$      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 $L_{1}(\mathbf{R})$      330
Approximate identity for $L_{1}(\mathbf{T})$      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 $\mathbf{R}$      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 $L_{p}[-\pi,\pi],p\geq1$      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 $N=2^{k}$      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 $L_{2}(\mathbf{R})$      366
Fourier integral formula      276 295—297
Fourier kernel, continuous $=sin\omega t/t$      111
Fourier kernel, discrete $=\frac{sin(n+1/2)t}{t/2}$      179
Fourier kernel, selector property      190 193 368
Fourier map      289—290
Fourier map, finite      266
Fourier map, for $L_{2}$ functions      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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