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What is an example of a continuous periodic function that is not the limit of any Fourier series? If not, is there an more or less elementary proof?

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    What makes you think there is one? Have you looked at the theorem on the convergence of Fourier series? What conditions can you violate? – Ross Millikan Nov 27 '20 at 19:52
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    Every periodic continuous function is the uniform limit of a sequence of Fourier series. Here there is a continuous function whose Fourier series $\lim_{N\to \infty}\lim_{M\to \infty}\sum_{n=-M}^N c_n e^{i nx}$ diverges at $0$ – reuns Nov 27 '20 at 20:24

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In part, there is some potential imprecision of language that confuses beginners here. I know it confused me. Plus, a surprising fact:

First, it has been known since at least since Fejer that every continuous function is a uniform-pointwise limit of finite Fourier series $\sum_{|m|\le N} c_{N,n} e^{2\pi inx}$.

More specifically, Fejer gave a formula for $c_{N,n}$ in terms of the Fourier coefficients $\widehat{f}(n)$ of the function $f$ itself...

The seeming paradox is that $c_{N,n}$ definitely cannot be $\widehat{f}(n)$ in general! That is, "the Fourier series" of $f$'s finite subsums are in general not the sequence of Fourier sums that converge uniformly to $f$. Crazy, seems to me.

paul garrett
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