Math 6610 • Analysis of Numerical Methods I

\textbf{a)} Show that the best uniform (\(L^\infty[-1,1]\)) approximation of an even
polynomial \(p_n(x)\) of (even) degree \(n\) by polynomials of degree \(\le n-2\) on
\([-1,1]\), is given by \(p_n(x)-c_nT_n(x)\), where \(T_n\) is the Tchebyshev polynomial
of degree \(n\), and \(c_n\) is an appropriately chosen constant. Hint: First find the
best approximation by polynomials of degree \(\le n-1\).

\textbf{b)} What are the zeroes of \(T_n\) and what are their significance in
interpolation. In particular, estimate the error of interpolation at Tchebyshev
nodes.