I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows:Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in $I=[-1/2,1/2]$.
(a) Prove the following reconstruction formula holds:$f(x)=\sum_{n=-\infty}^{\infty}f(n)K(x-n)$ where $K(y)=\sin{\pi y}/\pi y$.
(b) If $\lambda>1$, then $f(x)=\sum_{n=-\infty}^{\infty}1/\lambda f(n/\lambda)K_\lambda(x-n/\lambda)$ where $K_\lambda(y)=(\cos \pi y-\cos\pi\lambda y)/(\pi^2(\lambda-1)y^2)$.
(c) Prove that $\int_{-\infty}^\infty |f(x)|^2dx=\sum_{n=-\infty}^\infty|f(n)|^2$.
(a) is OK by expanding $\hat{f}$ by Fourier series in $I$ and using the Fourier inversion formula. The author gives hints: For part (b) show that $\hat{f}(\xi)=\chi(\xi)\sum_{-\infty}^\infty f(n)e^{-2\pi in\xi}$ where $\chi$ is 1 on $I$ and linear on $[-\lambda/2,-1/2]\cup[1/2,\lambda/2]$ and vanishes on $[-\lambda/2,\lambda/2]^c$.but I really don't know how to solve part b.
Is there any hint?
Thanks