Fourier transform supported on compact set

Question

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm convergence.

I know that $f$ must be continuous and going to $0$ at $\pm\infty$. The Fourier transform on $L^2$ is defined in a rather complicated way as a limit of Fourier transforms of functions in the Schwartz class. The right-hand side is an infinite sum (rather than the integral). How can we relate the two sides?

Answer 1

1. Expand $\hat f$ into a Fourier series on $[-\pi,\pi]$, that is $\hat f(y)=\sum_{n=-\infty}^\infty c_n e^{-iny}$. (I put $-$ in the exponential to get closer to the desired form; this does not change anything since $n$ runs over all integers anyway.)
2. Write $c_n $ as an integral of $\hat f(y)e^{iny}$ over $[-\pi,\pi]$.
3. Observe that the integral in 2, considered as an integral over $\mathbb R$, is the inverse Fourier transform. Recognize $f(n)$ in it.