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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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Eigenvalues 279—280 282 Eigenvalues, of 151 167 212 Eigenvalues, of Kronecker product 363 Eigenvalues, of LTI system 18 282 End padding operator 234 284 688 Equidistribution of arithmetic sequence 194 Error function 582 A36 Errors, for computation of 357 Errors, for computing sum with round off 793 Errors, for fast arithmetic 124 Errors, for FFT 301 Errors, for frame, detail coefficients 649—655 685—686 Errors, for least squares 75—76 84 Errors, for sampling theorems 492 497 499 507 Euclidean algorithm for gcd 227 Euler, gamma function 164 Euler, identity for sin, cos 1 67 Euler, Maclaurin sum formula 213 Even, function 62 64 247 Even, generalized function 397 455 Even, projection 245 Expectation integral 720 738 755—761 764 Expectation integral, for independent random variables 764 Expectation integral, for spectral density 720 Exponent notation 313 Exponent notation, via Kronecker products 339 Exponential operators 245 Extreme value principle 545 Factorial powers 440 Factorization, of 656 Factorization, of convolution operator 257 Factorization, of DFT matrix 314 329 338 Factorization, of DHT matrix 324 359 Factorization, of Q for filter bank 666 Fast arithmetic 113—114 124 Fast convolution 113 357 FBI filters 672 691' Fejer example of divergent Fourier series 57 85 Fejer kernel 78 474 Fermat theorem 236 FFT, Bluestein's scheme 358 FFT, Cooley — Tukey 295 349 FFT, decimation in frequency 299 318 338 FFT, decimation in time 296 316 322 331 FFT, for frames of movies 572 FFT, for spectral factorization 637 FFT, for spectrogram 705 FFT, FORTRAN code A27 FFT, Gauss discovery 70 295 360 FFT, impact 294—295 A1 FFT, in place 311 FFT, operations 295 323 332 353 358 FFT, Stockham's autosort 344 366 FFT, three loop algorithm 313 318 322 FFT, two loop algorithm 365 FFT, via DFT rules 296 299 353 FFT, via flow chart 348 A23 FFT, via Kronecker product 344 365—366 FFT, via Mason flow diagram 355 FFT, via matrix factorization 310 329 356 FFT, via recursive algorithm 301 350 351 FFT, via segments 352 FFT, via zipper identity 312 314 328 FFT, with precomputed sines 318 322 FHT, advantages 325 327 FHT, patent 325 Al FHT, three loop algorithm 326 FHT, via zipper identity 324 356 359 Filter bank 655 Filter bank, Fourier analysis 658 Filter bank, perfect reconstruction 661 689 Filter bank, using up, down sampling 656 Filter, FBI 672 691 Filter, for filter bank 659 Filter, for sampling 497 519 Filter, for shaping noise 721 Filter, high-, low-pass 281 511 659 Filter, translation 663 Filter, via convolution 497 Fletcher — Munsen contours 695—696 Floor function 124 200 229 383 467 683 703 717 731 FM synthesis of tone 711—717 FM synthesis of tone, parallel and cascade 734 Forward difference 118 440 480 FOURIER ii xiv xv Fourier coefficient 5 10 441 446 Fourier coefficient, rate of decay 193 441 Fourier series 5 173 Fourier series, convergence 39—48 75 77 Fourier series, for generalized functions 440—441 Fourier series, to solve PDEs 532 535 538 549 567 Fourier series, uniqueness 30 78 Fourier series, via Bernoulli functions 184 218 437 Fourier series, via differentiation 422 479 Fourier series, via integration 174 Fourier series, via Laurent series 185 Fourier series, via Poisson's formula 179 478 Fourier series, via Riemann sum 79 Fourier series, via rules 176—179 191—192 Fourier series, weak convergence 58 433 441 Fourier transform see “DFT” Fourier transform 129 114 146 Fourier transform 153 170 Fourier transform 413 Fourier transform 444 Fourier transform 746—752 Fourier transform 48 81 153 169 786 Fourier transform A1—A7 Fourier transform 132 421—425 Fourier transform 129—131 Fourier transform 134—147 413 Fourier transform 3 6 7 11 Fourier transform A14—A18 Fourier transform A1—A13 Fourier transform of operator 255 259 Fourier transform operator 240 243 Fourier transform operator, tag 251 Fourier — Poisson cube 31 36—37 205 Fourier, analysis and synthesis 3—11 15 73 86 Fourier, analysis and synthesis, validity 37—58 Fourier, and dimensional analysis 69 Fourier, and heat conduction 15 72 541 Fourier, big pixel image 510 Fourier, impact of work A1—A3 Fourier, quote iii Fourier, sketch xv 134 Fourier, spoken word 483 509 Fractional derivatives 154 Fragmentation of Ft 495 518 Frame for wavelet approximation 598 640 Frame for wavelet approximation, computation 606 608 649 652 Frame for wavelet approximation, illustrations 601 603 650 669 Fraunhofer approximation 565 569 571 588—589 Frequency, and pitch 694 Frequency, carrier-modulation 711 Frequency, for piano keyboard A37 Frequency, function for tune 723 Frequency, local 706 730 Frequency, of concert A 700 Frequency, of vibrating string 536 Frequency, units for 73 Frequency, via spectrogram 705 Frequency, via wavelet coefficients 596 Fresnel, approximation 555 Fresnel, convolution equation 558 Fresnel, function 165 420 560 588 Fresnel, function, discrete 214 Fresnel, integrals 165 215 420 466 FT-NMR spectrum 22 165 A2 Function, bandlimited 426 Function, CSG 376 Function, entire 517 Function, frequency 724 Function, generalized 58 368 378—379 524 Function, locally integrable 389 467 Function, of operator 244 272 277—278 281 283 Function, of random variable 758—760 789 Function, on 3—5 8 Function, probability density 739 Function, Schwartz 372—374 Function, slowly growing 376 Function, support-limited 426 Functional 369 Functional, continuous 451 Functional, fundamental 370 377—378 Functional, integral notation 372 378 Functional, linear 390 451 FWT, in place 676 FWT, via coefficients 606 FWT, via matrix factorization 675 FWT, via operators 645—646 Gauss, asteroid orbit 14 70 Gauss, discovery of FFT 71 295 360 Gauss, interpolation 8 14 360 Gauss, law of errors 771 794 Gauss, Poisson sum formula 179 Gauss, signature 669—670 Gauss, sums 215 Gaussian function 132 741 Gaussian function, for mollification and tapering 743 Gaussian laser beam 561 Gaussian laser beam, interfering 564 Gaussian laser beam, pointing 562 Gaussian laser beam, spreading 562 GCD 206 227 Generalized function 367—372 378 Generalized function, "continuity" 431 Generalized function, "values" 378 382 Generalized function, as functional 369—372 376 382 384 388 451 Generalized function, as ordinary function 58 376 379 Generalized function, as probability densities 739 Generalized function, as scaling function 616 623 Generalized function, as solution to PDE 524—525 528 541 559 587 Generalized function, bandlimited 426 Generalized function, closure 390 459 Generalized function, convolution 398—399 Generalized function, division 402—405 459 Generalized function, Fourier transform rules 413 Generalized function, integral notation 371 378 390—391 Generalized function, limits 427—439 Generalized function, multiplication 398—399 402 Generalized function, partial derivatives 438—439 525 Generalized function, periodic 440—448 Generalized function, special structure 408—410 427 441 459—464 Generalized function, table of Fourier transforms A4—A6 Generalized function, transformations 389—405 458 Generalized function, via CSG functions 378 Generalized function, via limit of Schwartz functions 450 482 Generalized function, via locally integrable functions 389 467 Generating function for, Bernoulli polynomials 225 Generating function for, Bessel functions 224 Generating function for, function on 101 Generating function for, Hermite function 167 Generating function for, Hermite polynomials 166 Geometric progression 25 65—66 228 GFT 450 Gibbs phenomenon 44 47—48 80 84 Gibbs phenomenon, for wavelets 624 687 Glissando of Risset 725 Goertzel algorithm 347 Grouping operator 659 Grouping rule 187 Haar wavelet 594 Haar wavelet, analysis 600 606 Haar wavelet, Fourier transform 596 Haar wavelet, scaling function 597 Haar wavelet, synthesis 594 606 Hanning window 229 Hartley transform 248 249 Hartley transform, advantages 248 255 263 279 Hartley transform, tag 251 Hartley transform, via Fourier transform 249 251 Hartley transform, via rules 263 265—266 285 Heat flow xiii 15 72 540—553 582—587 Heaviside function 116 131 380 424 Helmholtz, H. 707 Hermite functions 151 166 Hermite polynomials 151 160 Hermitian conjugation 62 Hermitian conjugation rule 136 Hermitian conjugation, operator 251 Hermitian conjugation, tag 252 Hilbert transform 266 471 Hilbert transform, for sampling theorem 519 Hilbert transform, tag 267 Hilbert transform, via analytic function 270 Hilbert transform, via Kramers — Kronig relations 269 Hilbert transform, via rules 267 288 Hipparchus — Ptolemy model 12 70 Horner's algorithm, for DFT 293 Horner's algorithm, for other tasks 345—347 Huygens synthesis of waves 555 Impulse response 282 481 Impulse response, automobile suspension 418 Impulse response, for ODE 419 470 471 Impulse response, for PDE see “Kernel” Impulse response, mass on spring 367—368 408 417 453 Independent random variables 764 Infinite product 615 Infinite series 1 5 Infinite series, of bandlimited functions 489 491 497 503 505 Infinite series, of generalized functions 431 435 Infinite series, of sinusoids 5 440 Infinite series, of solutions for PDE 16 532 549 567 Infinite series, weak convergence of 431 440 Initial "conditions" for PDEs 587 Integral notation 371 378 Interpolation, by piecewise linear function 145 234 485 Interpolation, by trigonometric polynomials 70 284 360 514 Interpolation, of bandlimited function 485 489 491 497 503 505 Interpolation, of Pallas orbit 14 70 360 Interpolation, using FFT 234 284 Inverse power function 387 395 Inversion rule 141 174 199 240 413 Involution 251 279 Isoperimetric inequality 226 235 Jumps in 46 55 123 184 Jumps in 184 479 Jumps in 406 Jumps in 47 84 624 Jumps in 46 55 Kasner's problem 125 Kernel, de la Vallee — Poussin 27 Kernel, diffraction 466 559 566 586 Kernel, diffusion 542 548 586 Kernel, Dirichlet 66 187 474 Kernel, Fejer 78 474 kernel, Poisson 65 Kernel, wave 528 532 575 576 Kramers — Kronig relations 269 289—290 Kronecker product 338 Kronecker product, algebraic properties 339 363 Kronecker product, eigenvalues 363 Kronecker product, rearrangement 341 Kronecker rule 174 Laplace function 135 Laplace's equation 525 591 laser beam 553 Laurent series, for ellipse 222 Laurent series, for Fourier series 185 187 218
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