Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Proof of Fejér's theorem
- 3 Weyl's equidistribution theorem
- 4 The Weierstrass polynomial approximation theorem
- 5 A second proof of Weierstrass's theorem
- 6 Hausdorff's moment problem
- 7 The importance of linearity
- 8 Compass and tides
- 9 The simplest convergence theorem
- 10 The rate of convergence
- 11 A nowhere differentiable function
- 12 Reactions
- 13 Monte Carlo methods
- 15 Pointwise convergence
- 16 Behaviour at points of discontinuity I
- 17 Behaviour at points of discontinuity II
- 18 A Fourier series divergent at a point
- 19 Pointwise convergence, the answer
- 20 The undisturbed damped oscillator does not explode
- 21 The disturbed damped linear oscillator does not explode
- 23 The linear damped oscillator with periodic input
- 27 Poisson summation
- 28 Dirichlet's problem
- 29 Potential theory with smoothness assumptions
- 30 An example of Hadamard
- 31 Potential theory without smoothness assumptions
- 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
- 43 Tchebychev and uniform approximation I
- 44 The existence of the best approximation
- 45 Tchebychev and uniform approximation II
- 46 Introduction to Fourier transforms
- 47 Change in the order of integration I
- 48 Change in the order of integration II
- 49 Fejér's theorem for Fourier transforms
- 50 Sums of independent random variables
- 51 Convolution
- 52 Convolution on T
- 53 Differentiation under the integral
- 54 Lord Kelvin
- 55 The heat equation
- 57 The age of the earth II
- 59 Weierstrass's proof of Weierstrass's theorem
- 60 The inversion formula
- 63 A second approach
- 64 The wave equation
- 71 The central limit theorem II
- 72 Stability and control
- 73 Instability
- 74 The Laplace transform
- 75 Deeper properties
- 76 Poles and stability
- 77 A simple time delay equation
- 79 Many dimensions
- 80 Sums of random vectors
- 81 A chi squared test
- 82 Haldane on fraud
- 86 Will a random walk return?
- 87 Will a Brownian motion return?
- 89 Will a Brownian motion tangle?
- 94 Why do we compute?
- 95 The diameter of stars
- 97 Fourier analysis on the roots of unity
- 99 How fast can we multiply?
- 102 A good code?
- 103 A little more group theory
- 104 Fourier analysis on finite Abelian groups
- 105 A formula of Euler
- 107 Primes in some arithmetical progressions
- 108 Extension from real to complex variable
- 109 Primes in general arithmetical progressions
- Appendixes A, B, G
- References
- Index
Preface
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Proof of Fejér's theorem
- 3 Weyl's equidistribution theorem
- 4 The Weierstrass polynomial approximation theorem
- 5 A second proof of Weierstrass's theorem
- 6 Hausdorff's moment problem
- 7 The importance of linearity
- 8 Compass and tides
- 9 The simplest convergence theorem
- 10 The rate of convergence
- 11 A nowhere differentiable function
- 12 Reactions
- 13 Monte Carlo methods
- 15 Pointwise convergence
- 16 Behaviour at points of discontinuity I
- 17 Behaviour at points of discontinuity II
- 18 A Fourier series divergent at a point
- 19 Pointwise convergence, the answer
- 20 The undisturbed damped oscillator does not explode
- 21 The disturbed damped linear oscillator does not explode
- 23 The linear damped oscillator with periodic input
- 27 Poisson summation
- 28 Dirichlet's problem
- 29 Potential theory with smoothness assumptions
- 30 An example of Hadamard
- 31 Potential theory without smoothness assumptions
- 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
- 43 Tchebychev and uniform approximation I
- 44 The existence of the best approximation
- 45 Tchebychev and uniform approximation II
- 46 Introduction to Fourier transforms
- 47 Change in the order of integration I
- 48 Change in the order of integration II
- 49 Fejér's theorem for Fourier transforms
- 50 Sums of independent random variables
- 51 Convolution
- 52 Convolution on T
- 53 Differentiation under the integral
- 54 Lord Kelvin
- 55 The heat equation
- 57 The age of the earth II
- 59 Weierstrass's proof of Weierstrass's theorem
- 60 The inversion formula
- 63 A second approach
- 64 The wave equation
- 71 The central limit theorem II
- 72 Stability and control
- 73 Instability
- 74 The Laplace transform
- 75 Deeper properties
- 76 Poles and stability
- 77 A simple time delay equation
- 79 Many dimensions
- 80 Sums of random vectors
- 81 A chi squared test
- 82 Haldane on fraud
- 86 Will a random walk return?
- 87 Will a Brownian motion return?
- 89 Will a Brownian motion tangle?
- 94 Why do we compute?
- 95 The diameter of stars
- 97 Fourier analysis on the roots of unity
- 99 How fast can we multiply?
- 102 A good code?
- 103 A little more group theory
- 104 Fourier analysis on finite Abelian groups
- 105 A formula of Euler
- 107 Primes in some arithmetical progressions
- 108 Extension from real to complex variable
- 109 Primes in general arithmetical progressions
- Appendixes A, B, G
- References
- Index
Summary
My book Fourier Analysis has no exercises and, in my view, is complete without them. However, exercises are useful both to the teacher of a course and to the student who wishes to learn by doing. This supplement provides such exercises (the exercises are grouped by chapter, although not all chapters have exercises).
The two remarks that follow are addressed to students using this book by themselves.
(1) I have tried to produce exercises and not problems. You should find them more in the nature of a hill top walk than a rock climbing expedition. I have marked some of the easier questions with a minus sign to prevent you searching for non-existent subtleties. Very occasionally part of a question is marked by a plus sign to indicate that further reflection may be required.
(2) Unless you intend to do all the questions, you should browse until you find a question or sequence of questions that interest you. You are more likely to pick up knowledge or technique from an exercise that interests you than from one that does not.
The references to other books and papers which occur from time to time are intended to encourage further reading, and not as a complete record of any indebtedness to other sources. The Cambridge Tripos examinations of various years have been the largest single source of questions, but experts will recognise the influence of the texts of Helson, Katznelson, Rogosinski, Dym and McKean and many others. Experts will also recall the verses of Kipling.
- Type
- Chapter
- Information
- Exercises in Fourier Analysis , pp. ix - xPublisher: Cambridge University PressPrint publication year: 1993