projection of band limit functions

Question

Define $$ P_{\Omega}f(x) = \frac{1}{2\pi}\int_{\mathbb{R}}e^{i\zeta x}\chi_{\{|\zeta|\leq \Omega\}}\hat{f}(\zeta)d\zeta $$ where $\hat{f}$ is the Fourier transform of $f$. Show that $P_{\Omega}^2 = P_{\Omega}$. I am trying to write out definitions and do computations. However, that makes a triple integral and I don't see how we cancel out terms here. I have shown that $P_{\Omega}$ is self-adjont.

Answer 1

Just notice that $P_\Omega=\mathcal{F}^{-1}\chi_{\{|\zeta|<\Omega\}}\mathcal{F}$ where $\mathcal{F}:L^2\rightarrow L^2$ is the Fourier transform and $\chi_{\{|\zeta|<\Omega\}}$ is a multiplication operator.