Principles of Fourier analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL
CRC Press, Taylor & Francis Group
[2017]
|
| Edition: | Second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (page 769) and index |
| Physical Description: | xvi, 788 pages Illustrationen 26 cm |
| ISBN: | 9781498734097 |
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MARC
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| 245 | 1 | 0 | |a Principles of Fourier analysis |c Kenneth B. Howell |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton, FL |b CRC Press, Taylor & Francis Group |c [2017] | |
| 300 | |a xvi, 788 pages |b Illustrationen |c 26 cm | ||
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| adam_text | Contents
Preface xiii
Sample Courses xv
I Preliminaries 1
1 The Starting Point 3
1.1 Fourier’s Bold Conjecture..................................................... 3
1.2 Mathematical Preliminaries and the Following Chapters........................ 5
Additional Exercises ............................................................. 6
2 Basic Terminology, Notation and Conventions 7
2.1 Numbers....................................................................... 7
2.2 Functions, Formulas and Variables............................................. 8
2.3 Operators and Transforms ................................................ 12
3 Basic Analysis I: Continuity and Smoothness 15
3.1 (Dis)Continuity............................................................ . 15
3.2 Differentiation............................................................ 22
3.3 Basic Manipulations and Smoothness ......................................... 25
3.4 Addenda.................................................................... 27
Additional Exercises ............................................................. 34
4 Basic Analysis II: Integration and Infinite Series 37
4.1 Integration ................................................................. 37
4.2 Infinite Series (Summations) .............................................. 41
Additional Exercises ....................................................... 48
5 Symmetry and Periodicity 49
5.1 Even and Odd Functions....................................................... 49
5.2 Periodic Functions .......................................................... 51
5.3 Sines and Cosines.......................................................... 53
Additional Exercises ............................................................. 56
6 Elementary Complex Analysis 57
6.1 Complex Numbers.......................................................... 57
6.2 Complex-Valued Functions..................................................... 59
6.3 The Complex Exponential ................................................. 61
6.4 Functions of a Complex Variable.............................................. 66
Additional Exercises ............................................................. 71
via
7 Functions of Several Variables 73
7.1 Basic Extensions ............................................................. 73
7.2 Single Integrals of Functions with Two Variables............................... 78
7.3 Double Integrals............................................................... 82
7.4 Addenda: Proving Theorems 7.7 and 7.9.......................................... 84
Additional Exercises ............................................................... 90
II Fourier Series 93
8 Heuristic Derivation of the Fourier Series Formulas 95
8.1 The Frequencies................................................................ 95
8.2 The Coefficients............................................................... 96
8.3 Summary........................................................................ 99
Additional Exercises .............................................................. 100
9 The Trigonometric Fourier Series 101
9.1 Defining the Trigonometric Fourier Series................................... 101
9.2 Computing the Fourier Coefficients ........................................... 107
9.3 Partial Sums and Graphing..................................................... 115
Additional Exercises .............................................................. 117
10 Fourier Series over Finite Intervals (Sine and Cosine Series) 121
10.1 The Basic Fourier Series..................................................... 121
10.2 The Fourier Sine Series...................................................... 123
10.3 The Fourier Cosine Series ................................................... 125
10.4 Using These Series........................................................... 126
Additional Exercises .............................................................. 127
11 Inner Products, Norms and Orthogonality 129
11.1 Inner Products............................................................... 129
11.2 The Norm of a Function ..................................................... 131
11.3 Orthogonal Sets of Functions................................................. 132
11.4 Orthogonal Function Expansions............................................... 134
11.5 The Schwarz Inequality for Inner Products.................................... 135
11.6 Bessel’s Inequality.......................................................... 137
Additional Exercises .............................................................. 141
12 The Complex Exponential Fourier Series 145
12.1 Derivation................................................................... 145
12.2 Notation and Terminology..................................................... 147
12.3 Computing the Coefficients................................................... 149
12.4 Partial Sums................................................................. 150
Additional Exercises .............................................................. 151
13 Convergence and Fourier’s Conjecture 155
13.1 Pointwise Convergence........................................................ 155
13.2 Uniform and Nonuniform Approximations........................................ 161
13.3 Convergence in Norm.......................................................... 169
13.4 The Sine and Cosine Series................................................. 173
Additional Exercises .............................................................. 175
ix
14 Convergence and Fourier’s Conjecture: The Proofs 179
14.1 Basic Theorem on Pointwise Convergence...................................... 179
14.2 Convergence for a Particular Saw Function................................... 186
14.3 Convergence for Arbitrary Saw Functions..................................... 195
14.4 A Divergent Fourier Series.................................................. 196
15 Derivatives and Integrals of Fourier Series 201
15.1 Differentiation of Fourier Series............................................201
15.2 Differentiability and Convergence............................................206
15.3 Integrating Periodic Functions and Fourier Series..........................210
15.4 Sine and Cosine Series.......................................................214
Additional Exercises ..............................................................216
16 Applications 219
16.1 The Heat Flow Problem.......................................................219
16.2 The Vibrating String Problem.................................................226
16.3 Functions Defined by Infinite Series.......................................234
16.4 Verifying the Heat Flow Problem Solution...................................243
Additional Exercises ..............................................................247
III Classical Fourier Transforms 249
17 Heuristic Derivation of the Classical Fourier Transform 251
17.1 Riemann Sums over the Entire Real Line.......................................251
17.2 The Derivation ..............................................................253
17.3 Summary......................................................................255
18 Integrals on Infinite Intervals 257
18.1 Absolutely Integrable Functions..............................................257
18.2 The Set of Absolutely Integrable Functions ................................261
18.3 Many Useful Facts ...........................................................261
18.4 Functions with Two Variables.................................................268
Additional Exercises ..............................................................276
19 The Fourier Integral Transforms 279
19.1 Definitions, Notation and Terminology........................................279
19.2 Near-Equivalence.............................................................281
19.3 Linearity....................................................................283
19.4 Invertibility................................................................284
19.5 Other Integral Formulas (A Warning)..........................................286
19.6 Some Properties of the Transformed Functions ................................287
Additional Exercises ..............................................................294
20 Classical Fourier Transforms and Classically Transformable Functions 297
20.1 The First Extension..........................................................298
20.2 The Set of Classically Transformable Functions.............................302
20.3 The Complete Classical Fourier Transforms....................................304
20.4 What Is and Is Not Classically Transformable? .............................308
20.5 Finite Duration and Finite Bandwidth Functions..............................310
20.6 More on Terminology, Notation and Conventions...............................313
Additional Exercises ............................................................. 314
X
21 Some Elementary Identities: Translation, Scaling and Conjugation 319
21.1 Translation ....................................................................319
21.2 Scaling .......................................................................327
21.3 Practical Transform Computing..................................................328
21.4 Complex Conjugation and Related Symmetries.....................................332
Additional Exercises ................................................................ 335
22 Differentiation and Fourier Transforms 339
22.1 The Differentiation Identities..................................................339
22.2 Rigorous Derivation of the Differential Identities ..........................346
22.3 Higher Order Differential Identities ...........................................349
22.4 Anti-Differentiation and Integral Identities...................................351
Additional Exercises .................................................................356
23 Gaussians and Gaussian-Like Functions 359
23.1 Basic Gaussians................................................................ 359
23.2 General Gaussians ..............................................................364
23.3 Gaussian-Like Functions.........................................................368
Additional Exercises .................................................................373
24 Convolution and Transforms of Products 375
24.1 Derivation of the Convolution Fonnula...........................................375
24.2 Basic Formulas and Properties of Convolution...................................377
24.3 Algebraic Properties ..........................................................379
24.4 Computing Convolutions.........................................................382
24.5 Existence, Smoothness and Derivatives of Convolutions..........................388
24.6 Convolution and Fourier Analysis...............................................392
Additional Exercises .................................................................395
25 Correlation, Square-Integrable Functions and the Fundamental Identity 399
25.1 Correlation.....................................................................399
25.2 Square-Integrable/Finite Energy Functions.......................................403
25.3 The Fundamental Identity........................................................412
Additional Exercises ............................................................... 416
26 Generalizing the Classical Theory: A Naive Approach 419
26.1 Delta Functions.................................................................419
26.2 Transforms of Periodic Functions................................................426
26.3 Arrays of Delta Functions.......................................................429
26.4 The Generalized Derivative.................................................... 432
Additional Exercises .................................................................444
27 Fourier Analysis in the Analysis of Systems 447
27.1 Linear, Shift-Invariant Systems.................................................447
27.2 Computing Outputs for LSI Systems..............................................454
Additional Exercises ............................................................... 461
28 Multi-Dimensional Fourier Transforms 463
28.1 Basic Definitions...............................................................463
28.2 Computing Multi-Dimensional Transforms ........................................466
Additional Exercises .................................................................470
xi
29 Identity Sequences 471
29.1 An Elementary Identity Sequence .............................................471
29.2 General Identity Sequences...................................................473
29.3 Gaussian Identity Sequences..................................................477
29.4 Verifying Identity Sequences.................................................481
29.5 An Application (with Exercises)..............................................485
29.6 Laplace Transforms as Fourier Transforms...................................487
Additional Exercises ..............................................................489
30 Gaussians as Test Functions and Proofs of Important Theorems 491
30.1 Testing for Equality with Gaussians........................................491
30.2 The Fundamental Theorem on Inverlibility...................................492
30.3 The Fourier Differential Identities..........................................495
30.4 The Fundamental and Convolution Identities of Fourier Analysis.............501
IV Generalized Functions and Fourier Transforms 509
31 A Starting Point for the Generalized Theory 511
31.1 Starting Points............................................................. 511
Additional Exercises ..............................................................514
32 Gaussian Test Functions 515
32.1 The Space of Gaussian Test Functions.......................................515
32.2 On Using the Space of Gaussian Test Functions..............................519
32.3 Other Test Function Spaces and a Confession................................521
32.4 More on Gaussian Test Functions..............................................522
32.5 Norms and Operational Continuity.............................................529
Additional Exercises ..............................................................535
33 Generalized Functions 537
33.1 Functionals..................................................................537
33.2 Generalized Functions .......................................................540
33.3 Basic Algebra of Generalized Functions.......................................547
33.4 Generalized Functions Based on Other Test Function Spaces..................553
33.5 Some Consequences of Functional Continuity...................................553
33.6 The Details of Functional Continuity........................................ 559
Additional Exercises ............................................................. 564
34 Sequences and Series of Generalized Functions 567
34.1 Sequences and Limits.........................................................567
34.2 Infinite Series (Summations).................................................574
34.3 A Little More on Delta Functions.............................................577
34.4 Arrays of Delta Functions....................................................579
Additional Exercises ..............................................................583
35 Basic Transforms of Generalized Fourier Analysis 587
35.1 Fourier Transforms...........................................................587
35.2 Generalized Scaling of the Variable ........................................ 592
35.3 Generalized Translation/Shifting.............................................597
35.4 The Generalized Derivative...................................................605
35.5 Transforms of Limits and Series..............................................613
xií
35.6 Adjoint-Defined Transforms in General........................................614
35.7 Generalized Complex Conjugation..............................................621
Additional Exercises ..........................................................623
36 Generalized Products, Convolutions and Definite Integrals 629
36.1 Multiplication and Convolution............................................630
36.2 Definite Integrals of Generalized Functions ..............................639
36.3 Appendix: On Defining Generalized Products and Convolutions...............643
Additional Exercises .............................................................646
37 Periodic Functions and Regular Arrays 649
37.1 Periodic Generalized Functions . .........................................649
37.2 Fourier Series for Periodic Generalized Functions.........................655
37.3 On Proving Theorem 37.5 .................................................... 663
Additional Exercises .............................................................671
38 Pole Functions and General Solutions to Simple Equations 673
38.1 Basics on Solving Simple Algebraic Equations .............................674
38.2 Homogeneous Equations with Polynomial Factors.............................677
38.3 Nonhomogeneous Equations with Polynomial Factors . ......................... 689
38.4 The Pole Functions....................................................... ■ . 693
38.5 Pole Functions in Transforms, Products and Solutions......................700
Additional Exercises ......................................................... · ■ 705
V The Discrete Theory 707
39 Periodic, Regular Arrays 709
39.1 The Index Period and Other Basic Notions..................................709
39.2 Fourier Series and Transforms of Periodic, Regular Arrays.................711
Additional Exercises .............................................................720
40 Sampling, Discrete Fourier Transforms and FFTs 721
40.1 Some General Conventions and Terminology .................................721
40.2 Sampling and the Discrete Approximation...................................722
40.3 The Discrete Approximation and Its Transforms............................ 725
40.4 The Discrete Fourier Transforms........................................... . 737
40.5 Discrete Transform Identities............................................... 741
40.6 Fast Fourier Transforms..................................................... 747
Additional Exercises .............................................................756
Tables, References and Answers 761
Table A.l: Fourier Transforms of Some Common Functions 763
Table A.2: Identities for the Fourier Transforms 767
References 769
Answers to Selected Exercises 771
Index
783
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| spelling | Howell, Kenneth B. Verfasser (DE-588)1120026431 aut Principles of Fourier analysis Kenneth B. Howell Second edition Boca Raton, FL CRC Press, Taylor & Francis Group [2017] xvi, 788 pages Illustrationen 26 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (page 769) and index Fourier analysis Textbooks Mathematical analysis Textbooks Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029698812&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Howell, Kenneth B. Principles of Fourier analysis Fourier analysis Textbooks Mathematical analysis Textbooks Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4023453-8 |
| title | Principles of Fourier analysis |
| title_auth | Principles of Fourier analysis |
| title_exact_search | Principles of Fourier analysis |
| title_full | Principles of Fourier analysis Kenneth B. Howell |
| title_fullStr | Principles of Fourier analysis Kenneth B. Howell |
| title_full_unstemmed | Principles of Fourier analysis Kenneth B. Howell |
| title_short | Principles of Fourier analysis |
| title_sort | principles of fourier analysis |
| topic | Fourier analysis Textbooks Mathematical analysis Textbooks Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Fourier analysis Textbooks Mathematical analysis Textbooks Harmonische Analyse |
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