Fourier transform formula

Question

How to prove that $f(x)=\sum_{n=-\infty}^\infty f(n)K(x-n)$ where $K(y)=\sin\pi y/\pi y$

Here, $f$ is moderate decrease and its fourier transform is supported in $[-1/2, 1/2]$.

I show that $\hat f(k)=\sum_{n=-\infty}^\infty f(n)e^{-2\pi ink}$ but no improvement...

Plz help

Answer 1

This formula is called Whittaker–Shannon interpolation formula.

Take inverse Fourier transform both sides of the formula $$\hat{f}(\xi)=\sum f(n) e^{-2\pi in\xi}\operatorname{rect}(\xi)$$

where $\operatorname{rect}(\xi)$ is rectangular function.