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Let $f\in L^1(\mathbb{T})$, and $\sum_{n}a_{n}e^{int}$ its Fourier series.

Fix a $t_{0}\in \mathbb{T}$. Suppose $\sum_{n}a_{n}e^{int}$ converges at $t_{0}$.

But if it is still possible that $\sum_{n}a_{n}e^{int}$ does not converge to $f(t)$ at $t_{0}$, then what terminology should we use for such a $t_{0}$?

Can we say that the Fourier series of $f$ converges at $t_{0}$?

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up vote 2 down vote accepted

It is possible: if you take an $f$ such that its Fourier series converges to it, and you change just one point, ie. $\tilde{f}(t) = f(t)$ except at some point $t_0$, the Fourier series of $\tilde{f}$ will still converge to $f$ and not to $\tilde{f}$.

In this case we say the series converges, but it does not converge to $\tilde{f}$.

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It is true that, for example, a $C^k$ function can be altered at a point, without changing the $L^2$ function "equivalence class", in this innocent/trivial way so that their Fourier series converge, but do not converge to the altered function. However, this is not the most interesting/important way in which Fourier series converge/not in various topologies. The far more important point is that $L^2$ convergence, the "natural convergence" of Fourier series, simply does not imply $C^o$ convergence (sup-norm). Constructive/useful ways to address this involve Sobolev spaces/norms, and such. – paul garrett Jul 27 '11 at 0:35

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