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85 results in Exercises in Fourier Analysis

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  • 10 - The rate of convergence

  • T. W. Körner, University of Cambridge
  • Book:
    Exercises in Fourier Analysis
    Published online:
    05 August 2012
    Print publication:
    19 August 1993, pp 41-45

  • Summary

    10.1 Jackson begins his book The Theory of Approximation with a reminiscence of the time when he did his PhD thesis with Landau.

    One day about twenty years ago I was admitted to the study of Professor Landau, seeking advice as to a subject for a thesis. After some preliminary inquiries as to my experience and preferences, he handed me a long sheet of paper, and directed me to take notes as he enumerated some dozen or fifteen topics in various fields of analysis and number theory, with a few words of explanation of each. He told me to think about them for a few days, and to select one of them, or any other problem of my own choosing, with the single reservation that I should not prove Fermat's theorem, an injunction which I have observed faithfully. Guided partly by natural inclination, perhaps, and partly by recollection of a course on methods of approximation which I had taken with Professor Bôcher a few years earlier, I committed myself to one of the topics which Landau had proposed, an investigation of the degree of approximation with which a given continuous function can be represented by a polynomial of given degree. When I reported my choice, he said meditatively, in words which I remember vividly in substance, if not perfectly as to idiom: ‘Das ist ein schönes Thema, ich beneide Sie um das Thema… Nein, ich beneide Sie nicht, aber es ist ein wunderschönes Thema!’ [That is a beautiful topic, I envy you that topic… No, I do not envy you, but it is a wonderful topic.]

  • 17 - Behaviour at points of discontinuity II

  • T. W. Körner, University of Cambridge
  • Book:
    Exercises in Fourier Analysis
    Published online:
    05 August 2012
    Print publication:
    19 August 1993, pp 59-62

  • 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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