Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
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1$\begingroup$ not converging in which topology? Please be specific. $\endgroup$ – Marc Palm Feb 4 '14 at 13:55
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The formulation seems specific enough to me. 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.

$\begingroup$ You should be able to remove your own comments. Isn't there a little x after it when you hover on it? $\endgroup$ – Gerald Edgar Feb 4 '14 at 16:35

1$\begingroup$ I wonder: is that capability (delete your own comments) only for registered users? $\endgroup$ – Gerald Edgar Feb 4 '14 at 16:50