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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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Step response 470 471 Stockham's autosort FFT 344 366 Strang, G. xii A1 Structure of this book xi Suitably regular 3 Sum of independent random variables 765 Sum of independent random variables, and central limit thoerem 771—779 Sum of independent random variables, mean, variance for 767 Sum of independent random variables, probability density for 766 Summation rule 204 209 258 262 Support-limited function 426 Support-limited wavelets 609 Symmetry, for boundary conditions 535 545 550—552 569 585 Symmetry, for deriving the Maxwell density 791 Symmetry, for solutions of PDEs 575 584 590 Symmetry, via Fourier transforms 62 64 66—67 477 Symmetry, via operators 239 254—256 277 Synthesis, using bandlimited functions 75—76 84 489 491 497 499 503 Synthesis, using cas 249 Synthesis, using complex exponentials 3 5 7 10 430 440 Synthesis, using sin, cos ix 67 247 Synthesis, using solutions of PDE 15 524 582 Synthesis, using wavelets 594 597 684 Tag notation 251—252 267 286 607 Tapered box 445—447 488 Tartaglia formula for roots of cubic 19 72 Taylor's formula 122 624 685 786 Temperature 15—16 540—553 Test functions 451 Thick coin 779 Timbre of tone 707 Transformation of music theme 724—725 Transformation shift rule see “Modulation rule” Translation invariant 17 282 Translation rule 136 140 177 199 413 Trapazoid rule, and Euler — Maclaurin formula 213 Trapazoid rule, and Fourier coefficients 234 Tree diagram for FFT 298 302 Triangle function 144 Truncated exponential 131 143 165 470 Truncated power function 382 Uncertainty relation 736 761 792 Unification of Fourier analysis, via Fourier — Poisson cube 36 Unification of Fourier analysis, via generalized functions 448—550 Unit gaussian function 132 Units for s, x, k, n 68 Universal constant 244 Upsampling see “Zero packing” Validity of Fourier's representation, for generalized functions 413 440—441 450 Validity of Fourier's representation, for ordinary functions 37—58 77 81—82 85 Validity of Fourier's representation, impact on mathematics A1—A3 Validity of wavelet representation 600 684 Variance 756 759 Variance 766—767 Vector operation for FFT 343 Velocity of traveling wave, group 589 Velocity of traveling wave, on string 528 Velocity of traveling wave, on water 591 Velocity of traveling wave, phase 566 589 Vibrating string 523 Vibrating string, bowed, plucked, struck 537 582 Vibrating string, discretized 580 Vibrating string, equation of motion 526 Vibrating string, frequency 536 Vibrating string, normal vibrational modes 536 Vibrating string, overtones 578 Vibrating string, send message 526 Vibrating string, shake to rest 577 Vibrating string, tone synthesis 539 710 731 Vibrating string, with stiffness 581 Vieta's formula 570 Water waves 554 591 Wave equation 523 527 Wave equation, conservation of energy 531 Wave equation, derivation 526 Wave equation, Fourier synthesis 524 532 Wave equation, initial conditions 527 531 587 Wave equation, kernel 528 532 575 576 Wave equation, polynomial solutions 574 Wave equation, symmetry 575 Wave equation, traveling solution 529—530 534 575—576 Wave equation, velocity 592 Wave equation, with boundary conditions 532 535 Wavelet 594 Wavelet transform 594 Wavelet, analysis equation 600 640 Wavelet, coefficients 609 Wavelet, continuity 683 Wavelet, Daubechies 610 614 Wavelet, frame-detail 598—604 Wavelet, Haar's prototype 594 Wavelet, having smoothness 616 683 Wavelet, mother-father 594 597 Wavelet, music score 597 600 674 Wavelet, samples 687 Wavelet, scaling function 597 609 Wavelet, support-limited 594 616 677 Wavelet, synthesis equation 594 684 Wavelet, via dilation equation 609 Wavelet, vs wave 593 Weak limit 428 Weak limit, for "continuity" 431 Weak limit, for central limit theorem 772 Weak limit, for derivative 430 Weak limit, for Fourier series 440—441 450 Weak limit, for initial conditions 587 Weak limit, for partial derivative 438 Weak limit, for sampling theorem 489 Weak limit, for solving dilation equation 614 Weak limit, for solving PDEs 587 Weak limit, transformations of 434 Weierstrass theorem 26 29 76 78 Weierstrass tone 731 Weyl's equidistribution theorem 195 Whittaker — Robinson flowchart for harmonic analysis 349 A23 Wiener's series for Fourier analysis 167 Window for DFT 229 705 Wirtinger's inequality 226 231 Young's double slit 560 588 Zero packing (upsampling) 187 201 232 257 261 656 Zipper identity 314 Zipper identity, as Kronecker product 344 Zipper identity, for FFT 312 327 Zipper identity, for FHT 323 359
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