Assume now we have $f(x)\in L^1([0,1])$, then we don't necessarily have the convergence of the partial sum of the Fourier series, moreover, by the theorem of kolmogorov, we can even have a.e. divergence of the partial sum.
Now my question is, for $f(x)\in L^1([0,1])$, denote the Fourier Transform as $\{a_n\}_{n=-\infty}^{\infty}$, and assume that the partial sum $S_n(x)=\sum_{k=-n}^{n}a_ke^{ikx}$ converges pointwise almost everywhere in $[0,1]$, then can we expect that the partial sum will converge back to the original function $f(x)$?