I am an undergrad studying Fourier series in the context of Riemann integrable functions.
I am wondering if $f\in L^2[-\pi,\pi]$ then does the Fourier series of $f$ converge to $f$ in the $L^2$ norm?
What I am having trouble with is, if $f$ is merely $L^2$, it may not be integrable, so how can I even compute the Fourier coefficients?
I know that the Fourier series is $ L^2$ convergent for Riemann integrable functions.
If the function is merely $ L^2$, do I know whether or not the function is Lesbesgue integrable?
Thanks
If $f\in L^2[-\pi,\pi]$, then $$ \int_{-\pi}^\pi f(x)e^{-inx}\mathrm dx=\langle f,e^{inx}\rangle $$ is well defined, it's the inner product of two functions in $L^2$! And yes the Fourier series converge to $f$ in $L^2$, as they are a complete system of functions.
Also, note that while this isn't necessary for your question, you do have integrability by Cauchy-Schwarz, $$ |\langle f,1\rangle|= \int_{-\pi}^{\pi}|f|\cdot 1\leq \sqrt{\int_{-\pi}^\pi |f|^2}\cdot\sqrt{2\pi} $$ and we have that $f\in L^1[-\pi,\pi]$.