My question is very simple. Suppose we have $S_n(x)$ the Fourier series of a function $F(x)$ from $\mathbb{R}\to \mathbb{R}$ and with period $T$. Does the series converge to $F(x)$ pointwise? Is the convergence uniform?
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It need not converge pointwise anywhere unless we assume something more of $F.$ – zhw. Jun 22 '17 at 15:56
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uhmm, no. The question of how 'good' the convergence of a Fourier series is is not a simple question. But what do you mean by "simple convergence"? – Thomas Jun 22 '17 at 15:57
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first of all excuse my bad english, i studied maths in french, i'll try to explain what i wanna do, i want to know if i can switch integral signe with sum sign in the serie, to do that the serie need to converge uniformally or normally that's what i wana know – Hptunjy Prjkeizg Jun 22 '17 at 15:58
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I have no doubt that it is reasonable to ask whether the Fourier series converges normally or uniformly, but as zhw has pointed out this depends on the properties you assume for $F$. It is rather easy to show uniform convergence for periodic $C^1$ functions, but that my be relaxed, e.g. to continuous with piecewise continuous derivative. For a proof you may want to have a look at this link: http://courses.mai.liu.se/GU/TATA57/Dokument/FourierSeries2.pdf – Thomas Jun 22 '17 at 16:06
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thanks alot ! i'll check it right now – Hptunjy Prjkeizg Jun 22 '17 at 16:08