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Uniqueness of fourier series

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that f is an integrable function (and suppose it's real valued) on the circle with c_n=0 for all n, where c_n stands for the coefficient of fourier series. Then f(p)=0 whenever f is continuous at the point p.


    2. Relevant equations



    3. The attempt at a solution
    assuming f is continuous at p=0, and supposing f(p)>0, the book (princeton lectures in analysis I) constructed trigonometric polynomials [tex]p_{k}(x)=(\epsilon+cos(x))^{k}[/tex] so that [tex]\int_{-\pi}^{\pi} f(x) p_{k}(x) dx[/tex] approaches infinity as k approaches infinity, contradicting the fact that the integral should* be zero. I have no idea why it should be zero. I thought c_n is defined to be [tex]\int_{-\pi}^{\pi} f(x) e^{-inx} dx[/tex] ? I do not see they are equivalent in some obvious way...what's the idea in it?
    Thanks a lot
     
    Last edited: Nov 17, 2009
  2. jcsd
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