Hermite and gauss type optimal quadratures for analytic functions

Abstract

For integrals\(\int\limits_{ - 1}^\iota {w(x)f(x)dx} \) with a non-negative weight functionw(x) and analyticf, we develop Hermite and Gauss type optimal quadratures over the Hilbert spaceH 2(Cr) of functions analytic in a circle. Our development of these optimal quadratures in very similar to that of classical Hermite and Gauss quadratures; the role played by fundamental polynomials in the classical theory is replaced here by certain «fundamental rational functions». We then show, by arguments similar to those used in the classical case, that a Gauss type optimal quadrature has positive weights an abscissas lying in (−1, 1).