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I'd really love to understand why does the fact that the series of the absolute Fourier coefficient converges ($\sum_{n \in \mathbb{Z}}|\widehat{f(n)}|<\infty$) for a function $f$, leads to the fact that the whole Fourier series, $\sum_{n \in \mathbb{Z}}\widehat{f(n)}e^{inx}$, uniformly converges.

Thanks a lot.

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up vote 5 down vote accepted

We have $\sup_{x\in\mathbb R}|\widehat f(n)e^{inx}|=|\widehat f(n)|$ so by Weierstrass M-test, the series $\sum_{n\in\mathbb Z}\widehat f(n)e^{inx}$ is normally convergent on the real line, hence uniformly convergent.

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