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Consider the Fourier series (in exponential form) generated by a function $f$ which is continuous on $[0,2\pi]$ and periodic with period $2\pi$ , say

$$f(x)\sim\sum_{n = - \infty }^{+ \infty }\alpha _{n}e^{inx}.$$

Assume also that the derivative $f '\in \mathbb{R}$ on $[0,2\pi]$.

a.) Prove that the series $\sum_{n = - \infty }^{+ \infty }n^{2}\left | \alpha _{n} \right |^{2}$ converges; then use the Cauchy-Schwarz inequality to deduce that $\sum_{n = - \infty }^{+ \infty }\left | \alpha _{n} \right |$ converges.

b.) From (a), deduce that the series $\sum_{n = - \infty }^{+ \infty }\alpha _{n}e^{inx}$ converges uniformly to a continuous sum function $g$ on $[0,2\pi]$. Then prove that $f=g$.

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Kindly tag it as homework if it so. –  user17762 Apr 24 '11 at 0:03
    
As usual: What have you tried? Where are you stuck? –  Eric Naslund Apr 24 '11 at 0:28
    
I'm having a difficult time showing showing that the series is piecewise differentiable so it can converge pointwise –  gumballjoe Apr 24 '11 at 13:45
    
the series i meant was the first one mentioned in letter a. Letter b i figured out last night so that one isnt necessary to be helped on. –  gumballjoe Apr 24 '11 at 14:12

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