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Kammler D.W. — First Course in Fourier Analysis
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Название: First Course in Fourier AnalysisАвтор: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: Second EditionГод издания: 2007Количество страниц: 798Добавлена в каталог: 13.04.2008Операции: Положить на полку |
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2 372 41 165 390 3 5 738 8 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 88 Analysis equation, for Dirac 429 Analysis equation, for periodic generalized 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 229 683 Central limit theorem 771—775 Central limit theorem, applications 775—779 793 Central limit theorem, for probability densities on 796 Central limit theorem, for probability densities on 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 741 Characteristic functions, for computing moments 757 Characteristic functions, for probability densities on 788 Characteristic functions, for probability densities on 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 390 396 459 Closure of 375—376 454 Closure of probability densities 787 Closure of symmetries 277 Coherent light 560 561 Comb function on 197 796 Comb function Ш on 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 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 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 279
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