Math 6420 • PDE

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