Convergent Fourier series of continuous function

Question

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 $S(x)\neq f(x)$).

Answer 1

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 $x$.

Answer 2

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)