Lagrange type errors for truncated Jacobi series (Q1100369)

Let \(s_ n\) denote the formal expansion of a function f in a Jacobi series truncated after \(n+1\) terms. For \(f\in C^{n+1}[-1,1]\) the uniform norm of \(f-s_ n\) is expressed in terms of the \((n+1)th\) derivative of f. This expression is precise when \(\max (\alpha,\beta)\geq\) and when \(-1<\alpha =\beta <\) with n odd. For other values of \(\alpha\) and \(\beta\) an asymptotic expression for the norm is derived. Comparisons are made with the minimax polynomial of degree no greater than n, for which it is known that \(\| f-p_ n\|_{\alpha}=(2^ n(n+1)!)^{-1}| f^{(n+1)}(\eta)|\) for some \(\eta\in (-1,1)\).