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Kammler D.W. — First Course in Fourier Analysis
Kammler D.W. — First Course in Fourier Analysis

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Название: First Course in Fourier Analysis

Автор: Kammler D.W.

Аннотация:

This unique book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDE's, probability, diffraction, musical tones, and wavelets. Providing unified development of (univariate) Fourier analysis for functions on R, T, Z, and P, the book also includes an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. It also uses the FT calculus and generalized functions to study the (univariate) wave equation, diffusion equation, and diffraction equation. In addition, fine points of the theory are developed. The book also demonstrates real-world applications of Fourier analysis in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of scientific professionals, including Mathematicians, Physicists, Chemists, Geologists, Electrical Engineers, Mechanical Engineers, and others.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 2007

Количество страниц: 798

Добавлена в каталог: 13.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$i\quad{\bf for} \sqrt{-1}$1      2
$\mathbb S$ for Schwartz functions      372
$\mathbb{C}$ for complex numbers      41 165
$\mathbb{R}$ for generalized functions      390
$\mathbb{R}$ for real numbers      3
$\mathbb{Z}$ for integers      5
${N}_{2}$ molecule      738
${\mathbb P}_{N}$ for polygon      8
${\mathbb T}_{p}$ for circle      4
Abel-Dirichlet argument      43
Absolutely integrable      48—50 57 81—82 90 122 126—127 164 169—171 179
Absolutely summable      38 40—41 127 210
Additive synthesis      ix
Additive synthesis, of solution to PDE      15 524
Additive synthesis, of tone      707—711 718—723 730—732
Alias      7 486 495 506 659 717 730
Almagast      12 699
Almost bandlimited      505
Almost periodic      88
Alternating flip      609 629 665 690
Analysis equation, for almost periodic $f$      88
Analysis equation, for Dirac $\delta$      429
Analysis equation, for periodic generalized $f$      446
Analysis equation, for wavelets      600 640
Analysis equation, of Fourier      3 6—7 10
Analysis equation, using sin, cos      12—13 66—68 247
Antiderivative rule      164
Antiderivative, of generalized function      390
Antiderivative, of probability density      379
Arginine spectrum      22 165
Arrow notation for      8 381
Audio signal      xiii 483—484 509 593 602 672 693—694
Autocorrelation      109 162 238
Automobile suspension      418
Averaging operator      72
B-spline      117 164 171 193 462 794
Bach, J.S.      725 735
Backward difference      230 see
Bandlimited function      426
Bandlimited function, almost      505
Bandlimited function, bound for derivatives      517
Bandlimited function, convolution of      511
Bandlimited function, discrete      512 515
Bandlimited function, for ear      485 728
Bandlimited function, for eye      510
Bandlimited function, limit of sequence      511
Bandlimited function, Maclaurin series for      486 517
Bandlimited function, multiplication by      477 487
Bandlimited function, recovery from filtered samples      499 503
Bandlimited function, recovery from samples      489 491 497 511
Bandlimited function, series for derivative      515
Bandlimited function, structure of      427 485
Bar operator      251
Bar operator, tag      252
Beethoven sonata      693
Bell curve      771 794
Bell tone      709 716 732
Benford's density for first digits      798
Bernoulli discretized string      580
Bernoulli functions      41—42 177 182 552
Bernoulli functions, and Eagle's method      184 479
Bernoulli functions, discrete      230
Bernoulli functions, for creating comb      432
Bernoulli functions, for Euler — Maclaurin formula      212 225
Bernoulli functions, for evaluating sums      221
Bernoulli functions, for synthesis of Fourier series      184
Bernoulli functions, properties      43—44 182—183 212 225
Bernstein's bound      516
Bessel functions      224 712—714 732—734 760
Bessel's inequality      26 77
Biorthoganality constraint      666 690
Bit reversal permutation      303 354
Bit reversal permutation, algorithms      305—307 310
Bit reversal permutation, operation count      354
Bit reversal permutation, via even-odd sorts      303 314—315 351
Bit reversal permutation, via Kronecker products      342 365
Bit reversal permutation, via perfect shuffles      314—315 342
Bluestein's chirp FFT      358
Bochner, S.      59 A1
Bohr, H.      88
Born and Wolf      553 573 587
Boundary conditions for PDE      535 545 550—552 569 585
Box function      130
Bracewell, R.      115
Bracewell, R., bit reversal      307—310
Bracewell, R., FHT      323 326 A1
Brass tone      710
Brass tone, analysis of Gibbs phenomenon      80
Brass tone, characterization of densities      749
Bunched samples      519
Buneman's tricks      308 319—320
Cardinal series      491 513
Cardinal series, basis functions      141 160 493 514—515
Cardinal series, convergence      492—493
Cardinal series, generalization      497 515
Carlson's theorem      57
Carrier frequency      711
Cascade algorithm      682
Cauchy — Schwartz inequality      491 507 521 745 756 762 792
Ceiling function $\lceil\quad\rceil$      229 683
Central limit theorem      771—775
Central limit theorem, applications      775—779 793
Central limit theorem, for probability densities on ${\mathbb P}_{N}$      796
Central limit theorem, for probability densities on ${\mathbb T}_{p}$      797
Centroid      74 756
Characteristic functions      741
Characteristic functions, Bochner's characterization      749
Characteristic functions, boundedness      748
Characteristic functions, continuity      748
Characteristic functions, convention for $2\pi$      741
Characteristic functions, for computing moments      757
Characteristic functions, for probability densities on ${\mathbb P}_{N}$      788
Characteristic functions, for probability densities on ${\mathbb T}_{p}$      797
Characteristic functions, for sums of random variables      765 767
Characteristic functions, from expectation integrals      759
Characteristic functions, periodic      752
Characteristic functions, products of      750
Characteristic functions, smoothness      786
Characteristic functions, via weak limit      748
Chirp signal      109 487 706 797
Chirp signal, for Bluestein's FFT      358
Chowning's rule      713—717 732
Chowning, J.      xiii 711 716
Circulant matrix      122
Closure of $\mathbb{G}$      390 396 459
Closure of $\mathbb{S}$      375—376 454
Closure of probability densities      787
Closure of symmetries      277
Coherent light      560 561
Comb function on ${\mathbb P}_{N}$      197 796
Comb function Ш on $\mathbb{R}$      383 393 432 437 448
Commensurate      88
Commuting diagram      37 205 256
Commuting operators      122
Complex exponentials, as eigenfunctions of LTI systems      18
Complex exponentials, on $\mathbb{R}, {\mathbb T}_{p}, {\mathbb Z}, {\mathbb P}_{N}}$      2 4 6 9
Compression of signal, via filter bank      668—670
Compression of signal, via interpolation      71 234
Concatenation of vectors      351 352 607—608 671 675
Concerning Harmonics      699
Conjugation      62
Conjugation rule      136 177 199 413
Conjugation, operator      251
Conjugation, operator, tag      292
Conservation of energy      24 531 544 565
Convention for 27r      63 741
Convolution      89—90
Convolution rule      103 143 170 177 199 257 286 413
Convolution rule, for Hartley transforms      265 286
Convolution rule, for Hilbert transforms      268 289
Convolution, algebraic properties      103 120 122 126—127 233
Convolution, and correlation      91
Convolution, and LTI systems      105 282 470 481
Convolution, and smoothness      107 123 126 128 401
Convolution, as filtering      497
Convolution, as smearing      107
Convolution, differentiation      106 121 126 401
Convolution, for solving ODEs      145 470 481
Convolution, for solving PDEs      528 534 542 548 559 567
Convolution, identity      104 118—120 168 196 400 408 see
Convolution, indirect      103 149 158
Convolution, of bandlimited functions      511
Convolution, of generalized functions      398 400 461
Convolution, of ordinary functions      89—90
Convolution, of periodic generalized functions      447
Convolution, of probability densities      110 123 150 751
Convolution, square root      105 232
Convolution, support      121
Convolution, translation      120 162
Convolution, via FFT      113 357
Convolution, via FHT      265
Convolution, via generating function      101
Convolution, via sliding strip      94
Cooley — Tukey      295 362 A1
Correlation      90 109 162
Cos operator      246 287
Cos transform      63 66 247 287
Cosine signal      511
CSG function      376
D'Alembert      523
D'Alembert formula      528 574 575
Dagger operator      251
Dagger operator, tag      252
Daubechies wavelets      631 639
Daubechies, I.      xiii 610 614 631
Decay of Fourier transform      153 170 193 376 786
Decibel      695
Decimation (downsampling) rule      188 204 258
Deconvolution      108 679
Definition chasing      391 471
Delta function on Pjy      196
density      see “Probability density”
Derivative operator      121 251 285
Derivative operator, tag      251
Derivative rule, for convolution      106 401
Derivative rule, for Fourier transform      141 169 413
Derivative rule, for generalized functions      377 405 413
Derivative rule, for Hartley transform      265 285
Derivative rule, for Hilbert transform      269
Derivative rule, of Leibnitz      107
Detail for wavelet approximation      600 640
Detail for wavelet approximation, computation of      606 608 649 652
DFT      11 291
DFT, eigenvalues      279
DFT, fast algorithms      see “FFT”
DFT, for problems of harmonic analysis      348
DFT, of real vector      276 352
DFT, table      A12
DFT, via geometric progression      198 227
DFT, via Horner's algorithm      293
DFT, via paper strips      294 349
DFT, via Poisson's relations      209
DFT, via rules      199—208
DFT, via summation      196
DHT      249
DHT, fast algorithms      see “FHT”
Dice      110 123 793
Dice, loaded      796
Dice, tetrahedral      780
Differential equation      163 281 408
Differential equation, for finding Fourier transform      132 164 420—425
Differential equation, homogeneous      408 416 460
Differential equation, inhomogeneous      146 367 408 410—412 417 419 461 470 477
Differential equation, via convolution      470
Differentiation of, generalized function      377 405
Differentiation of, piecewise smooth function      406—408 525
Diffraction      553
Diffraction equation      524
Diffraction equation, conservation of energy      565
Diffraction equation, derivation      553—558 587
Diffraction equation, Fourier synthesis      524 567 571—572
Diffraction equation, Fraunhofer approximation      565 569
Diffraction equation, initial condition      557 559 566 587
Diffraction equation, kernel      466 559 566 586
Diffraction equation, no extreme value principle      590
Diffraction equation, symmetry      590
Diffraction equation, with boundary conditions      569
Diffraction, from edge      588
Diffraction, from periodic source      568—571
Diffraction, from slits      560—561 588
Diffraction, of gaussian beam      ii 561—564
Diffusion equation      15 524
Diffusion equation, conservation of energy      544
Diffusion equation, derivation      540
Diffusion equation, extreme value principle      545
Diffusion equation, forced      546 585—586
Diffusion equation, Fourier synthesis      524 549 571—572
Diffusion equation, initial condition      541 548 587
Diffusion equation, kernel      542 548 586
Diffusion equation, polynomial solutions      544 584
Diffusion equation, smoothness of solutions      542 584
Diffusion equation, symmetry      584
Diffusion equation, with boundary conditions      545 550 551 552 585
Digitized sound file      xiii 483—484 508—509 593 602 672 693—694
Dilate of wavelet      594
Dilation equation      163 281 598 609
Dilation equation, and multiresolution analysis      599 645—646 650
Dilation equation, solution via infinite product      615 677
Dilation equation, symmetry      678
Dilation equation, uniqueness of solution      616 677
Dilation operator      258 283 656 659
Dilation rule      138 140 187 219 205—207 236 257 413
Dimensional analysis      69
Dipoles      429
Dirac delta      371 380 392—393
Dirac delta, and Eagle's method      421—425 479
Dirac delta, approximations      28 77—78 168 474
Dirac delta, as identity for convolution product      400
Dirac delta, as solution of dilation equation      610
Dirac delta, dilation property      392
Dirac delta, discrete      196
Dirac delta, not an ordinary function      168 453
Dirac delta, sifting property      392
Dirichlet conditions      57 77
Dirichlet kernel      66 187 219 474
Discrete      211
Discrete function      see also “Function on $\mathbb{Z}$
Discretization (sampling)      32 483—484
Dispersion, of heat      543
Dispersion, of light waves      566
Dispersion, of water waves      591
Distribution (tempered)      x 368 451
Distribution function      740
Distribution function, for max, min of random variables      790
Distribution function, for standard normal density      755 A33
Distribution function, via limits      784
Dual rule      260 280
Duality for Fourier analysis and synthesis      8 66
DuBois — Reymond      57
DWT      607 675
DWT, in place      676
DWT, operation count      608 692
DWT, using operators      646
DWT, via herringbone      608 646
Dyadic dilate      594
Eagle's method      184 422 479
Eavesdrop      512 520
Echo location      109
Eigenfunctions of operators      18 282
Eigenfunctions of operators, of ${\mathcal F}_{N}$      279
1 2 3 4
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