Fourier analysis and applications: filtering, numerical computation, wavelets
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
New York [u.a.]
Springer
1999
|
| Series: | Texts in applied mathematics
30 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Franz. übers. |
| Physical Description: | XVIII, 442 S. Ill., graph. Darst. |
| ISBN: | 0387984852 |
Staff View
MARC
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| 240 | 1 | 0 | |a Analyse de Fourier et applications |
| 245 | 1 | 0 | |a Fourier analysis and applications |b filtering, numerical computation, wavelets |c C. Gasquet ; P. Witomski |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XVIII, 442 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Texts in applied mathematics |v 30 | |
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Record in the Search Index
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Contents
Translator's Preface v
Preface to the French Edition vii
Chapter I Signals and Systems 1
Lesson 1 Signals and Systems 3
1.1 General considerations 3
1.2 Some elementary signals 6
1.3 Examples of systems 7
Lesson 2 Filters and Transfer Functions 11
2.1 Algebraic properties of systems 11
2.2 Continuity of a system 12
2.3 The filter and its transfer function 14
2.4 A standard analog filter: the RC cell 15
2.5 A first order discrete filter 18
Chapter II Periodic Signals 21
Lesson 3 Trigonometric Signals 23
3.1 Trigonometric polynomials 23
3.2 Representation in sines and cosines 24
3.3 Orthogonality 24
3.4 Exercises 26
Lesson 4 Periodic Signals and Fourier Series 27
4.1 The space L£(0,a) 27
4.2 The idea of approximation 29
xii Contents
4.3 Convergence of the approximation 31
4.4 Fourier coefficients of real, odd, and even functions 34
4.5 Formulary 35
4.6 Exercises 35
Lesson 5 Pointwise Representation 39
5.1 The Riemann Lebesgue theorem 39
5.2 Pointwise convergence? 40
5.3 Uniform convergence of Fourier series 45
5.4 Exercises 47
Lesson 6 Expanding a Function in an Orthogonal Basis 51
6.1 Fourier series expansions on a bounded interval 51
6.2 Expansion of a function in an orthogonal basis 53
6.3 Exercises 56
Lesson 7 Frequencies, Spectra, and Scales 57
7.1 Frequencies and spectra 57
7.2 Variations on the scale 59
7.3 Exercises 62
Chapter III The Discrete Fourier Transform and
Numerical Computations 63
Lesson 8 The Discrete Fourier Transform 65
8.1 Computing the Fourier coefficients 65
8.2 Some properties of the discrete Fourier transform 68
8.3 The Fourier transform of real data 71
8.4 A relation between the exact and approximate Fourier
coefficients 71
8.5 Exercises 73
Lesson 9 A Famous, Lightning Fast Algorithm 75
9.1 The Cooley Tukey algorithm 75
9.2 Evaluating the cost of the algorithm 77
9.3 The mirror permutation 78
9.4 A recursive program 80
9.5 Exercises 81
Lesson 10 Using the FFT for Numerical Computations 85
10.1 Computing a periodic convolution 85
10.2 Nonperiodic convolution 87
10.3 Computations on high order polynomials 88
10.4 Polynomial interpolation and the Chebyshev basis 90
10.5 Exercises 94
Contents xiii
Chapter IV The Lebesgue Integral 95
Lesson 11 From Riemann to Lebesgue 97
11.1 Some history 97
11.2 Another point of view 98
11.3 By way of transition 99
Lesson 12 Measuring Sets 101
12.1 Measurable sets and measure 101
12.2 Sets of measure zero 104
12.3 Measurable functions 105
12.4 Exercises 107
Lesson 13 Integrating Measurable Functions 111
13.1 Constructing the integral Ill
13.2 Elementary properties of the integral 113
13.3 The integral and sets of measure zero 115
13.4 Comparing the Riemann and Lebesgue integrals 116
13.5 Exercises 119
Lesson 14 Integral Calculus 121
14.1 Lebesgue's dominated convergence theorem 121
14.2 Integrals that depend on a parameter 122
14.3 Fubini's theorem 124
14.4 Changing variables in an integral 125
14.5 The indefinite Lebesgue integral and primitives 126
14.6 Exercises 128
Chapter V Spaces 131
Lesson 15 Function Spaces 133
15.1 Spaces of differentiable functions 133
15.2 Spaces of integrable functions 135
15.3 Inclusion and density 137
15.4 Exercises 139
Lesson 16 Hilbert Spaces 141
16.1 Definitions and geometric properties 141
16.2 Best approximation in a vector subspace 143
16.3 Orthogonal systems and Hilbert bases 146
16.4 Exercises 151
xiv Contents
Chapter VI Convolution and the Fourier
Transform of Functions 153
Lesson 17 The Fourier Transform of Integrable Functions 155
17.1 The Fourier transform on L^R) 155
17.2 Rules for computing with the Fourier transform 157
17.3 Some standard examples 159
17.4 Exercises 161
Lesson 18 The Inverse Fourier Transform 163
18.1 An inversion theorem for L:(R) 163
18.2 Some Fourier transforms obtained by the inversion formula 165
18.3 The principal value Fourier inversion formula 166
18.4 Exercises 169
Lesson 19 The Space ¥ (R) 171
19.1 Rapidly decreasing functions 171
19.2 The space S" (R) 172
19.3 The inverse Fourier transform on SP 174
19.4 Exercises 175
Lesson 20 The Convolution of Functions 177
20.1 Definitions and examples 177
20.2 Convolution in LX(E) 179
20.3 Convolution in LP(R) 180
20.4 Convolution of functions with limited support 183
20.5 Summary 184
20.6 Exercises 184
Lesson 21 Convolution, Derivation, and Regularization 187
21.1 Convolution and continuity 187
21.2 Convolution and derivation 187
21.3 Convolution and regularization 188
21.4 The convolution ¥ (R) * S" (R) 190
21.5 Exercises 191
Lesson 22 The Fourier Transform on L2(R) 193
22.1 Extension of the Fourier transform 193
22.2 Application to the computation of certain Fourier
transforms 196
22.3 The uncertainty principle 197
22.4 Exercises 199
Lesson 23 Convolution and the Fourier Transform 201
23.1 Convolution and the Fourier transform in L^R) 201
23.2 Convolution and the Fourier transform in L2(R) 203
Contents xv
23.3 Convolution and the Fourier transform: Summary 204
23.4 Exercises 206
Chapter VII Analog Filters 209
Lesson 24 Analog Filters Governed by a Differential
Equation 211
24.1 The case where the input and output are in y 211
24.2 Generalized solutions of the differential equation 213
24.3 The impulse response when deg P deg Q 213
24.4 Stability 215
24.5 Realizable systems 216
24.6 Gain and response time 217
24.7 The Routh criterion 218
24.8 Exercises 219
Lesson 25 Examples of Analog Filters 221
25.1 Revisiting the RC filter 221
25.2 The RLC circuit 222
25.3 Another second order filter: ^g"+g = f 225
25.4 Integrator and differentiator filters 227
25.5 The ideal low pass filter 228
25.6 The Butterworth filters 229
25.7 The general approximation problem 231
25.8 Exercises 232
Chapter VIII Distributions 233
Lesson 26 Where Functions Prove to Be Inadequate 235
26.1 The impulse in physics 235
26.2 Uncontrolled skid on impact 237
26.3 A new look derivation 239
26.4 The birth of a new theory 241
Lesson 27 What Is a Distribution? 243
27.1 The basic idea 243
27.2 The space 3 (R) of test functions 244
27.3 The definition of a distribution 245
27.4 Distributions as generalized functions 247
27.5 Exercises 249
Lesson 28 Elementary Operations on Distributions 251
28.1 Even, odd, and periodic distributions 251
28.2 Support of a distribution 253
xvi Contents
28.3 The product of a distribution and a function 254
28.4 The derivative of a distribution 255
28.5 Some new distributions 258
28.6 Exercises 261
Lesson 29 Convergence of a Sequence of Distributions 265
29.1 The limit of a sequence of distributions 265
29.2 Revisiting Dirac's impulse 266
29.3 Relations with the convergence of functions 267
29.4 Applications to the convergence of trigonometric series . . . 268
29.5 The Fourier series of Dirac's comb 270
29.6 Exercises 273
Lesson 30 Primitives of a Distribution 275
30.1 Distributions whose derivatives are zero 275
30.2 Primitives of a distribution 276
30.3 Exercises 278
Chapter IX Convolution and the Fourier
Transform of Distributions 281
Lesson 31 The Fourier Transform of Distributions 283
31.1 The spaced '(R) of tempered distributions 283
31.2 The Fourier transform on SP '(R) 287
31.3 Examples of Fourier transforms in ¥ '(R) 290
31.4 The space "(R) of distributions with compact support . . 291
31.5 The Fourier transform on "(R) 292
31.6 Formulary 294
31.7 Exercises 294
Lesson 32 Convolution of Distributions 297
32.1 The convolution of a distribution and a C°° function . 297
32.2 The convolution ' * 3 ' 301
32.3 The convolution '*y ' 303
32.4 The convolution S'+ *S'+ 304
32.5 The associativity of convolution 306
32.6 Exercises 308
Lesson 33 Convolution and the Fourier Transform of
Distributions 311
33.1 The Fourier transform and convolution Sf * Sf ' 311
33.2 The Fourier transform and convolution f' * Sf ' 312
33.3 The Fourier transform and convolution L2 * L2 313
33.4 The Hilbert transform 313
Contents xvii
33.5 The analytic signal associated with a real signal 314
33.6 Exercises 315
Chapter X Filters and Distributions 317
Lesson 34 Filters, Differential Equations, and Distributions 319
34.1 Filters revisited 319
34.2 Realizable, or causal, filters 321
34.3 Tempered solutions of linear differential equations 321
34.4 Exercises 324
Lesson 35 Realizable Filters and Differential Equations 325
35.1 Representation of the causal solution 325
35.2 Examples 327
35.3 Exercises 331
Chapter XI Sampling and Discrete Filters 333
Lesson 36 Periodic Distributions 335
36.1 The Fourier series of a locally integrable periodic function . 335
36.2 The Fourier series of a periodic distribution 337
36.3 The product of a periodic function and a periodic
distribution 340
36.4 Exercises 342
Lesson 37 Sampling Signals and Poisson's Formula 343
37.1 Poisson's formula in %' 344
37.2 Poisson's formula in Ll{WL) 345
37.3 Application to the study of the spectrum of a sampled signal 348
37.4 Application to accelerating the convergence of a Fourier
series 350
37.5 Exercises 351
Lesson 38 The Sampling Theorem and Shannon's Formula 353
38.1 Shannon's theorem 355
N
38.2 The case of a function f(t) = ]T c^e21"^1 356
n= N
38.3 Shannon's formula fails in Sf ' 357
38.4 The cardinal sine functions 357
38.5 Sampling and the numerical evaluation of a spectrum . . . 359
38.6 Exercises 361
Lesson 39 Discrete Filters and Convolution 365
xviii Contents
39.1 Discrete signals and filters 365
39.2 The convolution of two discrete signals 367
39.3 Cases where the two supports are not bounded 368
39.4 Summary 371
39.5 Causality and stability of a discrete filter 372
39.6 Exercises 374
Lesson 40 The 2 Transform and Discrete Filters 375
40.1 The 2 transform of a discrete signal 375
40.2 Applications to discrete filters 378
40.3 Exercises 381
Chapter XII Current Trends: Time—Frequency
Analysis 383
Lesson 41 The Windowed Fourier Transform 385
41.1 Limitations of standard Fourier analysis 385
41.2 Opening windows 386
41.3 Dennis Gabor's formulas 388
41.4 Comparing the methods of Fourier and Gabor 392
41.5 Exercises 394
Lesson 42 Wavelet Analysis 395
42.1 The basic idea: the accordion 395
42.2 The wavelet transform 397
42.3 Orthogonal wavelets 405
42.4 Multiresolution analysis of L2(R) 410
42.5 Multiresolution analysis and wavelet bases 413
42.6 Afternotes 428
42.7 Exercises 431
References 433
Index 437 |
| any_adam_object | 1 |
| author | Gasquet, Claude Witomski, Patrick |
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| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-09-10T00:55:18Z |
| institution | BVB |
| isbn | 0387984852 |
| language | English French |
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| spelling | Gasquet, Claude Verfasser aut Analyse de Fourier et applications Fourier analysis and applications filtering, numerical computation, wavelets C. Gasquet ; P. Witomski New York [u.a.] Springer 1999 XVIII, 442 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 30 Aus dem Franz. übers. ANÁLISE DE FOURIER larpcal Analyse numérique ram Filtres (mathématiques) ram Fourier, Analyse de - Problèmes et exercices ram Fourier, Analyse de ram Fourier-analyse gtt Ondelettes ram Fourier analysis Signalverarbeitung (DE-588)4054947-1 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 Signalverarbeitung (DE-588)4054947-1 s Witomski, Patrick Verfasser aut Texts in applied mathematics 30 (DE-604)BV002476038 30 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008338046&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gasquet, Claude Witomski, Patrick Fourier analysis and applications filtering, numerical computation, wavelets Texts in applied mathematics ANÁLISE DE FOURIER larpcal Analyse numérique ram Filtres (mathématiques) ram Fourier, Analyse de - Problèmes et exercices ram Fourier, Analyse de ram Fourier-analyse gtt Ondelettes ram Fourier analysis Signalverarbeitung (DE-588)4054947-1 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4054947-1 (DE-588)4023453-8 |
| title | Fourier analysis and applications filtering, numerical computation, wavelets |
| title_alt | Analyse de Fourier et applications |
| title_auth | Fourier analysis and applications filtering, numerical computation, wavelets |
| title_exact_search | Fourier analysis and applications filtering, numerical computation, wavelets |
| title_full | Fourier analysis and applications filtering, numerical computation, wavelets C. Gasquet ; P. Witomski |
| title_fullStr | Fourier analysis and applications filtering, numerical computation, wavelets C. Gasquet ; P. Witomski |
| title_full_unstemmed | Fourier analysis and applications filtering, numerical computation, wavelets C. Gasquet ; P. Witomski |
| title_short | Fourier analysis and applications |
| title_sort | fourier analysis and applications filtering numerical computation wavelets |
| title_sub | filtering, numerical computation, wavelets |
| topic | ANÁLISE DE FOURIER larpcal Analyse numérique ram Filtres (mathématiques) ram Fourier, Analyse de - Problèmes et exercices ram Fourier, Analyse de ram Fourier-analyse gtt Ondelettes ram Fourier analysis Signalverarbeitung (DE-588)4054947-1 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | ANÁLISE DE FOURIER Analyse numérique Filtres (mathématiques) Fourier, Analyse de - Problèmes et exercices Fourier, Analyse de Fourier-analyse Ondelettes Fourier analysis Signalverarbeitung Harmonische Analyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008338046&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT gasquetclaude analysedefourieretapplications AT witomskipatrick analysedefourieretapplications AT gasquetclaude fourieranalysisandapplicationsfilteringnumericalcomputationwavelets AT witomskipatrick fourieranalysisandapplicationsfilteringnumericalcomputationwavelets |