TITLE: An $L^1$ function whose Fourier series converges but not to itself
QUESTION [5 upvotes]: Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?

REPLY [2 votes]: Answered by alpha on MathOverflow: 

If the partial sums of the Fourier series converge to a function $g$ a.e., then so do the Cesaro means.  But these converge in the $L^1$-sense to the original function.  So the answer to your question is no.