If we assume $f$ is Riemann integrable and the Fourier series of $f$ is pointwise convergent, do we have...

Question

If we assume $f$ is Riemann integrable and the Fourier series of $f$ is pointwise convergent, do we have $f(x)=\sum_{n=-\infty}^{n=+\infty} \hat {f}(n) e^{2\pi i nx/L}$?

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Answer 1

The Cesaro means of the Fourier series converges to $f$ everywhere that $f$ is continuous, which is almost everywhere for a Riemann integrable function. So, if the Fourier series converges at a point of continuity of $f$, then it must converge to $f$. So under your assumptions, that would mean that $f$ equals the Fourier series at least at every point of continuity of $f$, which is almost everywhere.