Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$.
I have derived that $$\hat{f}(y)=\sum_{n=-\infty}^\infty f(n)1_{[-\pi,\pi]}(y)e^{-iny}$$ in $L^2$ convergence.
Let $K(y)=\dfrac{\sin(\pi y)}{\pi y}$. If I take the termwise inverse Fourier transform, I would get $$f(x)=\sum_{n=-\infty}^\infty f(n)K(x-n)$$ But it is illegal to take the termwise inverse Fourier transform in an infinite series. How can I show that the series on the right-hand side converges for (almost) every $x$, and converges to $f(x)$?
(Note: This is related to this (not yet correctly answered) bounty question.)