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Fourier Analysis

Book Description

Fourier analysis is a subject that was born in physics but grew up in mathematics. Now it is part of the standard repertoire for mathematicians, physicists and engineers. In most books, this diversity of interest is often ignored, but here Dr Körner has provided a shop-window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy, and electrical engineering. Each application is placed in perspective by a short essay. The prerequisites are few (the reader with knowledge of second or third year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. In short, this stimulating account will be welcomed by all who like to read about more than the bare bones of a subject. For them this will be a meaty guide to Fourier analysis.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. Part I: Fourier Series
    1. 1 Introduction
    2. 2 Proof of Fejér’s Theorem
    3. 3 Weyl’s Equidistribution Theorem
    4. 4 The Weierstrass Polynomial Approximation Theorem
    5. 5 A Second Proof of Weierstrass’s Theorem
    6. 6 Hausdorff’s Moment Problem
    7. 7 The Importance of Linearity
    8. 8 Compass and Tides
    9. 9 The Simplest Convergence Theorem
    10. 10 The Rate of Convergence
    11. 11 A Nowhere Differentiable Function
    12. 12 Reactions
    13. 13 Monte Carlo Methods
    14. 14 Mathematical Brownian Motion
    15. 15 Pointwise Convergence
    16. 16 Behaviour at Points of Discontinuity I
    17. 17 Behaviour at Points of Discontinuity II
    18. 18 A Fourier Series Divergent at a Point
    19. 19 Pointwise Convergence, the Answer
  8. Part II: Some Differential Equations
    1. 20 The Undisturbed Damped Oscillator Does not Explode
    2. 21 The Disturbed Damped Linear Oscillator Does not Explode
    3. 22 Transients
    4. 23 The Linear Damped Oscillator with Periodic Input
    5. 24 A Non-Linear Oscillator I
    6. 25 A Non-Linear Oscillator II
    7. 26 A Non-Linear Oscillator III
    8. 27 Poisson Summation
    9. 28 Dirichlet’s Problem for the Disc
    10. 29 Potential Theory with Smoothness Assumptions
    11. 30 An Example of Hadamard
    12. 31 Potential Theory Without Smoothness Assumptions
  9. Part III: Orthogonal Series
    1. 32 Mean Square Approximation I
    2. 33 Mean Square Approximation II
    3. 34 Mean Square Convergence
    4. 35 The Isoperimetric Problem I
    5. 36 The Isoperimetric Problem II
    6. 37 The Sturm–Liouville Equation I
    7. 38 Liouville
    8. 39 The Sturm–Liouville Equation II
    9. 40 Orthogonal Polynomials
    10. 41 Gaussian Quadrature
    11. 42 Linkages
    12. 43 Tchebychev and Uniform Approximation I
    13. 44 The Existence of the Best Approximation
    14. 45 Tchebychev and Uniform Approximation II
  10. Part IV: Fourier Transforms
    1. 46 Introduction
    2. 47 Change in the Order of Integration I
    3. 48 Change in the Order of Integration II
    4. 49 Fejér’s Theorem for Fourier Transforms
    5. 50 Sums of Independent Random Variables
    6. 51 Convolution
    7. 52 Convolution on T
    8. 53 Differentiation Under the Integral
    9. 54 Lord Kelvin
    10. 55 The Heat Equation
    11. 56 The Age of the Earth I
    12. 57 The Age of the Earth II
    13. 58 The Age of the Earth III
    14. 59 Weierstrass’s Proof of Weierstrass’s Theorem
    15. 60 The Inversion Formula
    16. 61 Simple Discontinuities
    17. 62 Heat Flow in a Semi-Infinite Rod
    18. 63 A Second Approach
    19. 64 The Wave Equation
    20. 65 The Transatlantic Cable I
    21. 66 The Transatlantic Cable II
    22. 67 Uniqueness for the Heat Equation I
    23. 68 Uniqueness for the Heat Equation II
    24. 69 The Law of Errors
    25. 70 The Central Limit Theorem I
    26. 71 The Central Limit Theorem II
  11. Part V: Further Developments
    1. 72 Stability and Control
    2. 73 Instability
    3. 74 The Laplace Transform
    4. 75 Deeper Properties
    5. 76 Poles and Stability
    6. 77 A Simple Time Delay Equation
    7. 78 An Exception to a Rule
    8. 79 Many Dimensions
    9. 80 Sums of Random Vectors
    10. 81 A Chi Squared Test
    11. 82 Haldane on Fraud
    12. 83 An Example of Outstanding Statistical Treatment I
    13. 84 An Example of Outstanding Statistical Treatment II
    14. 85 An Example of Outstanding Statistical Treatment III
    15. 86 Will a Random Walk Return?
    16. 87 Will a Brownian Motion Return?
    17. 88 Analytic Maps of Brownian Motion
    18. 89 Will a Brownian Motion Tangle?
    19. 90 La Famille Picard va á Monte Carlo
  12. Part VI: Other Directions
    1. 91 The Future of Mathematics Viewed from 1800
    2. 92 Who was Fourier? I
    3. 93 Who was Fourier? II
    4. 94 Why do we Compute?
    5. 95 The Diameter of Stars
    6. 96 What do we Compute?
    7. 97 Fourier Analysis on the Roots of Unity
    8. 98 How do we Compute?
    9. 99 How Fast can we Multiply?
    10. 100 What Makes a Good Code?
    11. 101 A Little Group Theory
    12. 102 A Good Code?
    13. 103 A Little More Group Theory
    14. 104 Fourier Analysis on Finite Abelian Groups
    15. 105 A Formula of Euler
    16. 106 An Idea of Dirichlet
    17. 107 Primes in Some Arithmetical Progressions
    18. 108 Extension from Real to Complex Variable
    19. 109 Primes in General Arithmetical Progressions
    20. 110 A Word from our Founder
  13. Appendix A: The Circle T
  14. Appendix B: Continuous Function on Closed Bounded Sets
  15. Appendix C: Weakening Hypotheses
  16. Appendix D: Ode to a Galvanometer
  17. Appendix E: The Principle of the Argument
  18. Appendix F: Chase the Constant
  19. Appendix G: Are Share Prices in Brownian Motion?
  20. Index