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Pointwise convergence of integral of Fourier series

  1. May 17, 2010 #1
    1. The problem statement, all variables and given/known data
    If [tex]f(x)[/tex] is a piecewise-continuous function in [tex][-L,L][/tex], show that its indefinite integral [tex]F(x) = \int_{-L}^x f(s) ds[/tex] has a full Fourier series that converges pointwise.

    2. Relevant equations
    Full Fourier series: [tex]f(x)=\frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos (\frac{n \pi }{L}x) + B_n \sin (\frac{n \pi}{L}x) [/tex]

    Definition: [tex]\sum_{n=1}^\infty f_n (x) [/tex] converges to [tex]f(x)[/tex] pointwise in [tex](a,b)[/tex] if for each [tex]a<x<b[/tex] we have
    [tex]\Big| f(x) - \displaystyle{\sum_{n=1}^\infty f_n (x)} \Big| \to 0 [/tex] as [tex]N\to\infty[/tex].

    3. The attempt at a solution
    I think I need to somehow justify integrating term-by-term, but am not sure how to proceed. Any ideas?
  2. jcsd
  3. May 17, 2010 #2
    If you want to integrate term by term, you need uniform convergence.
    Haven't really looked at this, so not saying that term by term integration is the solution here.
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