Cambridge A First Course In Fourier Analysis

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Selected Applications......Page 11
To the Student......Page 13
Synopsis......Page 15
To the Instructor......Page 17
Acknowledgments......Page 18
Introduction......Page 21
Functions on R......Page 23
Functions on Tp......Page 24
Functions on Z......Page 26
Functions on PN......Page 28
Summary......Page 31
The Hipparchus–Ptolemy model of planetary motion......Page 32
Gauss and the orbits of the asteroids......Page 34
Fourier and the .ow of heat......Page 35
Fourier’s representation and LTI systems......Page 36
Schoenberg’s derivation of the Tartaglia–Cardan formulas......Page 39
Fourier transforms and spectroscopy......Page 41
The Parseval identities......Page 43
Orthogonality relations for the periodic complex exponentials......Page 44
Bessel’s inequality......Page 45
The Weierstrass approximation theorem......Page 46
A proof of Plancherel’s identity for functions on Tp......Page 50
Introduction......Page 51
Periodization by p -summation......Page 52
The Poisson relations......Page 53
The Fourier–Poisson cube......Page 56
Introduction......Page 57
Absolutely summable functions on Z......Page 58
Continuous piecewise smooth functions on Tp......Page 59
The sawtooth singularity function on T1......Page 61
The Gibbs phenomenon for w0......Page 64
Piecewise smooth functions on Tp......Page 65
Smooth functions on R with small regular tails......Page 68
Singularity functions on R......Page 71
Piecewise smooth functions on R with small regular tails......Page 75
Extending the domain of validity......Page 77
Further reading......Page 79
Exercises......Page 81
2.1 Formal definitions of…......Page 109
Direct evaluation......Page 111
The sum of scaled translates......Page 112
The sliding strip method......Page 114
Generating functions......Page 121
The Fourier transform of…......Page 122
Algebraic structure......Page 123
Translation invariance......Page 125
Differentiation of…......Page 126
Convolution as smearing......Page 127
Echo location......Page 129
Convolution and probability......Page 130
Convolution and arithmetic......Page 132
Further reading......Page 135
Exercises......Page 136
Introduction......Page 149
The box function......Page 150
The truncated decaying exponential......Page 151
The unit gaussian......Page 152
Introduction......Page 154
Linearity......Page 155
Translation and modulation......Page 156
Dilation......Page 158
Inversion......Page 161
Convolution and multiplication......Page 163
Summary......Page 166
Evaluation of integrals and sums......Page 167
Evaluation of convolution products......Page 169
The Hermite functions......Page 171
Smoothness and rates of decay......Page 173
Further reading......Page 175
Exercises......Page 176
Introduction......Page 193
Direct integration......Page 194
Elementary rules......Page 196
Poisson’s relation......Page 199
Bernoulli functions and Eagle’s method......Page 202
Laurent series......Page 205
Dilation and grouping rules......Page 207
Evaluation of sums and integrals......Page 210
The polygon function......Page 211
Rates of decay......Page 213
Equidistribution of arithmetic sequences......Page 214
Direct summation......Page 216
Basic rules......Page 219
Dilation......Page 225
Poisson’s relations......Page 229
The Euler–Maclaurin sum formula......Page 232
The discrete Fresnel function......Page 234
Further reading......Page 236
Exercises......Page 237
Operators applied to functions on PN......Page 259
Blanket hypotheses......Page 261
Powers of F......Page 263
The normalized exponential transform operators......Page 265
The normalized cosine transform and sine transform operators......Page 266
The normalized Hartley transform operators......Page 268
Connections......Page 269
The bar and dagger operators......Page 271
Symmetric functions......Page 273
Symmetric operators......Page 274
The basic definition......Page 275
Algebraic properties......Page 279
5.5 Rules for Hartley transforms......Page 283
Defining relations......Page 286
Operator identities......Page 287
The Kramers–Kronig relations......Page 289
Further reading......Page 291
Exercises......Page 292
Introduction......Page 311
Horner’s algorithm for computing the DFT......Page 313
How big is 4N2?......Page 314
The announcement of a fast algorithm for the DFT......Page 315
Decimation-in-time......Page 316
Decimation-in-frequency......Page 319
Recursive algorithms......Page 321
Introduction......Page 323
A naive algorithm......Page 324
The reverse carry algorithm......Page 325
The Bracewell–Buneman algorithm......Page 327
Introduction......Page 330
The zipper identity......Page 331
Sparse matrix factorization of F......Page 333
The action of B2m......Page 334
An FFT algorithm......Page 335
An alternative FFT algorithm......Page 337
Precomputation of…......Page 338
Application of Q4M......Page 340
The zipper identity and factorization......Page 343
Application of T4M using precomputed…......Page 344
The zipper identity for FMP......Page 347
An FFT......Page 349
The permutation…......Page 352
The permutation…......Page 353
The Kronecker product......Page 358
Rearrangement of Kronecker products......Page 361
Parallel and vector operations......Page 363
Stockham’s autosort FFT......Page 364
Exercises......Page 365
Introduction......Page 387
Functions and functionals......Page 389
Schwartz functions......Page 392
Functionals for generalized functions......Page 396
Introduction......Page 399
The comb III......Page 403
The functions…......Page 405
Summary......Page 408
Introduction......Page 409
The linear space G......Page 410
Translate, dilate, derivative, and Fourier transform......Page 411
Reflection and conjugation......Page 416
Multiplication and convolution......Page 418
Differentiation rules......Page 425
Derivatives of piecewise smooth functions with jumps......Page 426
Solving differential equations......Page 431
Fourier transform rules......Page 433
Basic Fourier transforms......Page 434
Support- and bandlimited generalized functions......Page 446
Introduction......Page 447
The limit concept......Page 448
Transformation of limits......Page 454
Fourier series......Page 460
The analysis equation......Page 465
Convolution of p-periodic generalized functions......Page 467
Discrete Fourier transforms......Page 468
Functionals on S......Page 470
Other test functions......Page 471
Further reading......Page 472
Exercises......Page 473
Introduction......Page 503
Shannon’s hypothesis......Page 504
A weakly convergent series......Page 507
The cardinal series......Page 511
Fragmentation of…......Page 515
Filters......Page 517
Samples from one filter......Page 518
The Papoulis generalization......Page 519
8.4 Approximation of almost bandlimited functions......Page 525
Further reading......Page 528
Exercises......Page 529
9.1 Introduction......Page 543
A physical context: Plane vibration of a taut string......Page 546
The wave equation on R......Page 547
The wave equation on Tp......Page 551
Each point on the string vibrates with the frequency......Page 556
A physical context: Heat .ow along a long rod......Page 560
The diffusion equation on R......Page 561
The diffusion equation on Tp......Page 568
A physical context: Diffraction of a laser beam......Page 573
The diffraction equation on R......Page 579
The diffraction equation on Tp......Page 586
9.5 Fast computation of frames for movies......Page 591
Further reading......Page 593
Exercises......Page 594
10.1 The Haar wavelets......Page 613
Interpretation of F[m,k]......Page 615
Arbitrarily good approximation......Page 617
Successive approximation......Page 622
Coded approximation......Page 624
The dilation equation......Page 629
Smoothness constraints......Page 636
Order of approximation......Page 640
Orthogonality constraints......Page 645
Daubechies wavelets......Page 651
Coefficients for frames and details......Page 660
The operators…......Page 664
Samples for frames and details......Page 669
The operators…......Page 672
Introduction......Page 675
Factorization of…......Page 676
Fourier analysis of a filter bank......Page 678
Perfect reconstruction .lter banks......Page 681
Compression and reconstruction......Page 688
Further reading......Page 693
Exercises......Page 694
Introduction......Page 713
Perception of pitch and loudness......Page 714
Ohm’s law......Page 716
Scales......Page 717
Musical notation......Page 720
Introduction......Page 722
The computation......Page 723
Slowly varying frequencies......Page 725
Amplitude envelopes......Page 727
Synthesis of a bell tone......Page 729
Synthesis of a brass tone......Page 730
Introduction......Page 731
The spectral decomposition......Page 732
Dynamic spectral enrichment......Page 735
White noise......Page 738
Filtered noise......Page 741
Introduction......Page 743
Transformation of frequency functions......Page 744
Risset’s endless glissando......Page 745
Further reading......Page 747
Exercises......Page 748
Introduction......Page 757
Generalized probability densities......Page 759
Gaussian molli.cation and tapering......Page 761
Fundamental inequalities......Page 764
F is bounded and continuous......Page 766
Bochner’s characterization of F......Page 769
Products of characteristic functions......Page 770
Periodic characteristic functions......Page 771
Probability integrals......Page 773
Expectation integrals......Page 775
Functions of a random variable......Page 778
The uncertainty relation......Page 781
Sums of independent random variables......Page 784
The ubiquitous bell curve......Page 791
The law of errors......Page 795
Exercises......Page 800
Appendix 1 The impact of Fourier analysis......Page 819
Appendix 2 Functions and their Fourier transforms......Page 822
Appendix 3 The Fourier transform calculus......Page 832
Appendix 4 Operators and their Fourier transforms......Page 837
Appendix 5 The Whittaker–Robinson flow chart for harmonic analysis......Page 841
Appendix 6 FORTRAN code for a radix 2 FFT......Page 845
Appendix 7 The standard normal probability distribution......Page 851
Appendix 8 Frequencies of the piano keyboard......Page 855
Index......Page 857