Math 6420 • PDE

The Legendre polynomials are given by
\[
P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n,
\]
and are eigenfunctions of the differential operator \(Lu=-((1-x^2)u')'\) on the interval
\(-1<x<1\).
\begin{enumerate}
\item[(a)] Find the eigenvalue \(\lambda_n\) corresponding to the eigenfunction \(P_n(x)\).
\item[(b)] In what sense, if any, are the functions \(\{P_n(x)\}\) complete?
\item[(c)] Prove or disprove that \(L\) is a positive definite operator.
\end{enumerate}