Is Gauss quadrature optimal for analytic functions?

M. A. Kowalski, Arthur G. Werschulz & Henryk Wozniakowski
We consider the problem of optimal quadratures for integrands f: [ -1 , 1 ] → R which have an analytic extension f to an open disk D, of radius r about the origin such that |f| ≤ 1 on Dr. 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...
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