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
17 - Behaviour at points of discontinuity II
Published online by Cambridge University Press: 05 August 2012
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
Summary
17.1 Let Kn be the Fejér kernel discussed in Chapter 2.
(i) Show that, if κ(n) → ∞ as n → ∞, then ∫|s|≥κ(n)/nKn(s)ds → 0 as n → ∞. (We shall be interested in slowly diverging κ(n) such as κ(n) = log(n + 2).)
(ii) Show that if h is the saw tooth function of Chapter 17 then
Conclude that nothing resembling the Gibbs phenomenon can occur for σn(h, t). Extend this observation to cover well-behaved functions with only a finite number of discontinuities. (If you actually graph σn(h, t) you may find that, in spite of what we have proved, σn(h,) is not a very good copy of h for small n, being a bit ‘round shouldered’ near π. Small degree trigonometric polynomials just do not look very much like discontinuous functions.)
17.2 As C. H. Su has pointed out to me, the result of Theorem 17.1 can be interpreted by using a technique from boundary layer theory (and elsewhere) and ‘blowing up’ the independent variable x near π. More precisely, instead of considering Sn(h, x) we introduce a new variable y = n(π − x).
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- Exercises in Fourier Analysis , pp. 59 - 62Publisher: Cambridge University PressPrint publication year: 1993
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