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[Nxb]

I can't solve a problem in Stein's book Fourier analysis. There's a stronger version of Tauber's theorem by Littlewood that if $\sum c_n$ is Abel summable or Cesaro summable to s and $c_n=O(1/n)$, then $\sum c_n$ converges to s. Since the Fourier series of a function is Abel summable to the points of continuity, if f is an integrable function that satisfies the Fourier coefficients $a_n=O(1/|n|)$, then the Fourier series converges at all points of continuity. But i can't explain why if f is continuous, then the Fourier series converges to f uniformly.

Edited by Nxb, 26-06-2016 - 09:43.