Math 6420 • PDE

Let \(P_n(x)\), \(n=0,1,\cdots\), be the Legendre polynomials,
\[
  P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n,
\]
which are also eigenfunctions for the linear differential operator
\[
  Lu=-\frac{d}{dx}\left((1-x^2)\frac{du}{dx}\right)
\]
on the interval \(-1\le x\le 1\).
\begin{enumerate}
\item[(a)] Determine the eigenvalue associated with \(P_n\)

\item[(b)] Show that the Legendre polynomials are orthogonal.

\item[(c)] Suppose \(f(x)\) is square integrable on the interval \(-1<x<1\). Use that the Legendre
polynomials are orthogonal to prove that
\[
  g(x)=\sum_{n=0}^{\infty}\alpha_nP_n(x)
\]
where
\[
  \alpha_n=
  \frac{\int_{-1}^{1}f(x)P_n(x)\,dx}{\int_{-1}^{1}P_n^2(x)\,dx},
\]
is also square integrable.
\end{enumerate}