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Fourier Series - proving function is continuous

  1. Jun 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Let f be an integrable, periodic function whose Fourier coefficients satisfy [itex] \sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty [/itex]. Prove that f is continuous.


    2. Relevant equations

    Looking at my notes, the only relevent things i have for this question (i think) are Bessel's inequality (but that requires f to be continuous....), the Riemann-Lebesgue Theorem, Parseval's equality and also the decay and regularity of fourier series. The statement that the uniform limit of a sequence of continuous functions is continuous, maybe I can use this?


    3. The attempt at a solution

    I thought this question would be simple, but I'm finding that its not. I have tried 2 methods:
    1) [itex]\sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty [/itex] implies [itex] n^6 |\hat{f}(n)|^2 \rightarrow 0 [/itex] hence [itex]|\hat{f}(n)| \rightarrow 0 [/itex]. I need to show that f is continuous i.e. the converse of the Riemann Lebesgue Lemma. How would I proceed from here?
    2) using L^2 convergence theorem (parseval's), [itex]\sum_{-\infty}^{\infty} |c_n|^2 \< \infty [/itex] hence there exists function g such that [itex]\hat{g}(n)=c_n=n^3 |\hat{f}(n)| [/itex]. Is this the route I should proceed?

    The hypothesis also states that f is integrable, but i dont know how to use this here! Im also thinking of the M test, but dont know if I can apply it to a sum like this... Any help will be appreciated!
     
  2. jcsd
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