# An L1 function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?

• not converging in which topology? Please be specific. – Marc Palm Feb 4 '14 at 13:55

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.