Is Gauss quadrature optimal for analytic functions? (Q1062222)

We consider the problem of optimal quadratures for integrands f: [- 1,1]\(\to {\mathbb{R}}\) which have an analytic extension \(\bar f\) to an open disk \(D_ r\) of radius r about the origin such that \(| \bar f| \leq 1\) on \(\bar D_ r\). If \(r=1\), we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an ''infinite'' penalty for using Gauss quadrature. On the other hand, if \(r>1\), Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.