Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether one can find a continuous function $f$ with the property that its Fourier series is convergent everywhere but not to $f$ (in other words for each $x$ the partial sums $S_N(x)$ converge to $S(x)$ and there are some points $x$ such that $S_N(x)$ converges to $S(x)\neq f(x)$).
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When you talk about the convergence of the Fourier series of $f$, is it pointwise convergence, or $L^2$ convergence? – Alex M. May 16 '16 at 13:48
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Convergent everywhere=pointwise for each point. – truebaran May 16 '16 at 13:52
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If the partial sums $S_N(x)$ converge to $S(x)$, then the Cesaro sums $$ \sigma_N(x)=\frac1N\sum_{k=1}^NS_N(x) $$ also converge to $S(x)$. But the Cesaro sums of a continuous function $f$ converge uniformly to $f$, so that $S(x)=f(x)$ for all $x$.
Julián Aguirre
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The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s, resolved positively in 1966 by Lennart Carleson in $L^2$ and generalized by Richard Hunt to $L^p$ for any $p > 1$. This result is known as the Carleson–Hunt theorem.
This article may help.
Andrey Kolmogorov, as a student at the age of 19, constructed an example of a function in $L^1$ whose Fourier series diverges almost everywhere (Original Article)
alexjo
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The Carleson–Hunt theorem states that if $f\in L^p$, for $p>1$, the Fourier series exists and converges to $f$ almost everywhere. Kolmogorov constructed a funcion $f\in L^1$ for which the Fourier series exists (it satisfies the weak Dirichlet's condition) but diverges almost everywhere. So this may be the example you are looking for. – alexjo May 16 '16 at 14:19
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I know those results: what I'm looking for is a continuous function $f$ with the property that its Fourier series converges everywhere to something (let us call this sum $S(x)$) but this something is different from $f$. Taking into account Carleson theorem obviously the set of points $x$ for which $S(x) \neq f(x)$ is of measure zero. I'm not interested in divergent series, I'm intersted in comparing the value of this sum with initial function $f$. Is it now clear? – truebaran May 16 '16 at 14:25
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@JuliánAguirre 's answer is correct.
But that's true only if the partial sum $S_N(x)$ converges at every $x$.
I guess what OP wanted to ask was if one could find a function $f$ whose Fourier series $S(x)$ doesn't converge to $f$ at some point. Note that this needs not to mean that $S(x)$ converges at every point. If $S(x)$ diverges at some point, then it clearly doesn't converge to $f$.
So I want to post a useful and relevant fact here.
Fact: There exist a continuous function $f$ whose Fourier series doesn't converge to $f$ on a null set (i.e. set of zero measure).
Here is a constructive proof from an anonym, c.f. Example of a function whose Fourier Series fails to converge at One point.
And here is an abstract proof from Wikipedia, c.f. https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Pointwise_convergence.
Sam Wong
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