Math 6420 • PDE

\begin{enumerate}
\item[(a)] Show that the functions \(\{e^{inx}\}_{n=-\infty}^{\infty}\) form an orthogonal
set in \(L^2[0,2\pi]\).
\item[(b)] Suppose \(f(x)\in L^2[0,2\pi]\). Let
\(\alpha_n=\int_0^{2\pi}f(x)e^{inx}\,dx\). Prove that
\(\{\alpha_n\}_{n=-\infty}^{\infty}\) is convergent in \(\ell^2\) and also that
\(\sum_{n=-\infty}^{\infty}\alpha_ne^{inx}\) is convergent in \(L^2[0,2\pi]\). What does
\(\sum_{n=-\infty}^{\infty}\alpha_ne^{inx}\) converge to?
\item[(c)] Suppose \(f(x)\) and \(g(x)\) are \(2\pi\) periodic functions and square integrable on
the interval \(0<x<2\pi\). State and prove the Convolution Theorem for these functions, that
is, find the Fourier coefficients of the function
\(h(x)=\int_0^{2\pi} f(x-t)g(t)\,dt\) in terms of the Fourier coefficients of \(f(x)\) and
\(g(x)\).
\end{enumerate}