Math 6420 • PDE

Remark: For this problem you may use without proof that
\[
  \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\mu x}\,d\mu=\delta(x).
\]
\begin{enumerate}
\item[(a)] Suppose \(f(x)\) has the Fourier transform \(F(\mu)\). Evaluate
\[
  \int_{-\infty}^{\infty} f(x)\overline{f(x-k)}\,dx
\]
in terms of \(F(\mu)\).
\item[(b)] Use this identity to prove that the set
\(\{\operatorname{sinc}(t-k)\}_{k=-\infty}^{\infty}\) forms an orthonormal set.
\end{enumerate}