Math 6210 • Real Analysis

Let \(H_n\) be the linear span in \(L^2([-\pi ,\pi ])\) of the set \(\{e^{ikt}\}_{k=-n}^n\). The orthogonal projection \(P_n\) of \(L^2([-\pi ,\pi ])\) onto \(H_n\) may be written in the form \[ P_n f(x)=h_n*f(x)=\frac {1}{2\pi }\int _{-\pi }^{\pi }h_n(x-t)f(t)\,dt. \] Find the function \(h_n\) and prove that it has this property.