Fourier analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore
Cambridge University Press
2022
|
| Ausgabe: | Reprinted with foreword |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiv, 591 Seiten Illustrationen, Diagramme |
| ISBN: | 9781009230056 |
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| 245 | 1 | 0 | |a Fourier analysis |c T. W. Körner (University of Cambridge) ; with a foreword by Terence Tao (University of California, Los Angeles) |
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| 264 | 1 | |a Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore |b Cambridge University Press |c 2022 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents Foreword by Terence Tao Preface Part I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Introduction Proof of Fejér’s theorem Weyl’s equidistribution theorem The Weierstrass polynomial approximation theorem A second proof of Weierstrass’s theorem Hausdorff’s moment problem The importance of linearity Compass and tides The simplest convergence theorem The rate of convergence A nowhere differentiable function Reactions Monte Carlo methods Mathematical Brownian motion Pointwise convergence Behaviour at points of discontinuity I Behaviour at points of discontinuity II A Fourier series divergent at a point Pointwise convergence, the answer Part II 20 Fourier Series Some Differential Equations The undisturbed damped oscillator does not explode Page xi xiii 1 3 6 11 15 19 21 24 28 32 35 38 42 46 50 56 59 62 67 74 77 79 vii
Contents viii 21 22 23 24 25 26 27 28 29 30 31 The disturbed damped linear oscillator does not explode Transients The linear damped oscillator with periodic input A non-linear oscillator I A non-linear oscillator II A non-linear oscillator III Poisson summation Dirichlet’s problem for the disc Potential theory with smoothness assumptions An example of Hadamard Potential theory without smoothness assumptions Part III Orthogonal Series 32 Mean square approximation I 33 Mean square approximation II 34 Mean square convergence 35 The isoperimetric problem I 36 The isoperimetric problem II 37 The Sturm-Liouville equation I 38 Liouville 39 The Sturm-Liouville equation II 40 Orthogonal polynomials 41 Gaussian quadrature 42 Linkages 43 Tchebychev and uniform approximation I 44 The existence of the best approximation 45 Tchebychev and uniform approximation II Part IV 46 47 48 49 50 51 52 53 54 Fourier Transforms Introduction Change in the order of integration I Change in the order of integration II Fejér’s theorem for Fourier transforms Sums of independent random variables Convolution Convolution on T Differentiation under the integral Lord Kelvin 83 88 93 99 104 113 116 121 124 131 134 143 145 150 155 159 166 170 175 179 185 191 197 201 207 212 219 221 226 230 240 245 253 259 265 270
Contents 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 The heat equation The age of the earth I The age of the earth II The age of the earth III Weierstrass’s proof of Weierstrass’s theorem The inversion formula Simple discontinuities Heat flow in a semi-infinite rod A second approach The wave equation The transatlantic cable I The transatlantic cable II Uniqueness for the heat equation I Uniqueness for the heat equation II The law of errors The central limit theorem I The central limit theorem II Part V 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Further Developments Stability and control Instability The Laplace transform Deeper properties Poles and stability A simple time delay equation An exception to a rule Many dimensions Sums of random vectors A chi squared test Haldane on fraud An example of outstanding statistical treatment I An example of outstanding statistical treatment II An example of outstanding statistical treatment III Will a random walk return? Will a Brownian motion return? Analytic maps of Brownian motion Will a Brownian motion tangle? La Famille Picard va á Monte Carlo ix 274 282 285 289 292 295 300 308 315 324 332 335 338 344 347 349 357 363 365 368 372 379 386 395 403 407 413 418 425 429 434 436 443 451 455 461 467
x Part VI Contents Other Directions 471 91 The future of mathematics viewed from 1800 92 Who was Fourier? I 93 Who was Fourier? II 94 Why do we compute? 95 The diameter of stars 96 What do we compute? 97 Fourier analysis on the roots of unity 98 How do we compute? 99 How fast can we multiply? 100 What makes a good code? 101 A little group theory 102 A good code? 103 A little more group theory 104 Fourier analysis on finite Abelian groups 105 A formula of Euler 106 An idea of Dirichlet 107 Primes in some arithmetical progressions 108 Extension from real to complex variable 109 Primes in general arithmetical progressions 110 A word from our founder Appendix A: The circle T Appendix B: Continuous function on closed bounded sets Appendix C: Weakening hypotheses Appendix D: Ode to a galvanometer Appendix E: The principle of the argument Appendix F: Chase the constant Appendix G: Are share prices in Brownian motion? 473 475 478 481 484 488 491 497 500 503 506 509 513 519 525 532 539 546 552 558 560 563 565 575 577 580 581 Index 585
|
| adam_txt |
Contents Foreword by Terence Tao Preface Part I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Introduction Proof of Fejér’s theorem Weyl’s equidistribution theorem The Weierstrass polynomial approximation theorem A second proof of Weierstrass’s theorem Hausdorff’s moment problem The importance of linearity Compass and tides The simplest convergence theorem The rate of convergence A nowhere differentiable function Reactions Monte Carlo methods Mathematical Brownian motion Pointwise convergence Behaviour at points of discontinuity I Behaviour at points of discontinuity II A Fourier series divergent at a point Pointwise convergence, the answer Part II 20 Fourier Series Some Differential Equations The undisturbed damped oscillator does not explode Page xi xiii 1 3 6 11 15 19 21 24 28 32 35 38 42 46 50 56 59 62 67 74 77 79 vii
Contents viii 21 22 23 24 25 26 27 28 29 30 31 The disturbed damped linear oscillator does not explode Transients The linear damped oscillator with periodic input A non-linear oscillator I A non-linear oscillator II A non-linear oscillator III Poisson summation Dirichlet’s problem for the disc Potential theory with smoothness assumptions An example of Hadamard Potential theory without smoothness assumptions Part III Orthogonal Series 32 Mean square approximation I 33 Mean square approximation II 34 Mean square convergence 35 The isoperimetric problem I 36 The isoperimetric problem II 37 The Sturm-Liouville equation I 38 Liouville 39 The Sturm-Liouville equation II 40 Orthogonal polynomials 41 Gaussian quadrature 42 Linkages 43 Tchebychev and uniform approximation I 44 The existence of the best approximation 45 Tchebychev and uniform approximation II Part IV 46 47 48 49 50 51 52 53 54 Fourier Transforms Introduction Change in the order of integration I Change in the order of integration II Fejér’s theorem for Fourier transforms Sums of independent random variables Convolution Convolution on T Differentiation under the integral Lord Kelvin 83 88 93 99 104 113 116 121 124 131 134 143 145 150 155 159 166 170 175 179 185 191 197 201 207 212 219 221 226 230 240 245 253 259 265 270
Contents 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 The heat equation The age of the earth I The age of the earth II The age of the earth III Weierstrass’s proof of Weierstrass’s theorem The inversion formula Simple discontinuities Heat flow in a semi-infinite rod A second approach The wave equation The transatlantic cable I The transatlantic cable II Uniqueness for the heat equation I Uniqueness for the heat equation II The law of errors The central limit theorem I The central limit theorem II Part V 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Further Developments Stability and control Instability The Laplace transform Deeper properties Poles and stability A simple time delay equation An exception to a rule Many dimensions Sums of random vectors A chi squared test Haldane on fraud An example of outstanding statistical treatment I An example of outstanding statistical treatment II An example of outstanding statistical treatment III Will a random walk return? Will a Brownian motion return? Analytic maps of Brownian motion Will a Brownian motion tangle? La Famille Picard va á Monte Carlo ix 274 282 285 289 292 295 300 308 315 324 332 335 338 344 347 349 357 363 365 368 372 379 386 395 403 407 413 418 425 429 434 436 443 451 455 461 467
x Part VI Contents Other Directions 471 91 The future of mathematics viewed from 1800 92 Who was Fourier? I 93 Who was Fourier? II 94 Why do we compute? 95 The diameter of stars 96 What do we compute? 97 Fourier analysis on the roots of unity 98 How do we compute? 99 How fast can we multiply? 100 What makes a good code? 101 A little group theory 102 A good code? 103 A little more group theory 104 Fourier analysis on finite Abelian groups 105 A formula of Euler 106 An idea of Dirichlet 107 Primes in some arithmetical progressions 108 Extension from real to complex variable 109 Primes in general arithmetical progressions 110 A word from our founder Appendix A: The circle T Appendix B: Continuous function on closed bounded sets Appendix C: Weakening hypotheses Appendix D: Ode to a galvanometer Appendix E: The principle of the argument Appendix F: Chase the constant Appendix G: Are share prices in Brownian motion? 473 475 478 481 484 488 491 497 500 503 506 509 513 519 525 532 539 546 552 558 560 563 565 575 577 580 581 Index 585 |
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| spelling | Körner, T. W. 1946- Verfasser (DE-588)120225379 aut Fourier analysis T. W. Körner (University of Cambridge) ; with a foreword by Terence Tao (University of California, Los Angeles) Reprinted with foreword Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore Cambridge University Press 2022 xiv, 591 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Fourier-Transformation (DE-588)4018014-1 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Fourier-Reihe (DE-588)4155109-6 gnd rswk-swf 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content Harmonische Analyse (DE-588)4023453-8 s Fourier-Reihe (DE-588)4155109-6 s Fourier-Transformation (DE-588)4018014-1 s DE-604 Tao, Terence 1975- (DE-588)132190370 wpr Erscheint auch als Online-Ausgabe 978-1-009-23006-3 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=033620522&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Körner, T. W. 1946- Fourier analysis Fourier-Transformation (DE-588)4018014-1 gnd Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd |
| subject_GND | (DE-588)4018014-1 (DE-588)4023453-8 (DE-588)4155109-6 (DE-588)4143389-0 |
| title | Fourier analysis |
| title_auth | Fourier analysis |
| title_exact_search | Fourier analysis |
| title_exact_search_txtP | Fourier analysis |
| title_full | Fourier analysis T. W. Körner (University of Cambridge) ; with a foreword by Terence Tao (University of California, Los Angeles) |
| title_fullStr | Fourier analysis T. W. Körner (University of Cambridge) ; with a foreword by Terence Tao (University of California, Los Angeles) |
| title_full_unstemmed | Fourier analysis T. W. Körner (University of Cambridge) ; with a foreword by Terence Tao (University of California, Los Angeles) |
| title_short | Fourier analysis |
| title_sort | fourier analysis |
| topic | Fourier-Transformation (DE-588)4018014-1 gnd Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd |
| topic_facet | Fourier-Transformation Harmonische Analyse Fourier-Reihe Aufgabensammlung |
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