Degree optimal average quadrature rules for the generalized Hermite weight function

For the practical estimation of the error of Gauss quadrature rules Gauss-Kronrod rules are widely used; but, it is well known that for the generalized Hermite weight function, ωα(x) = |x|2α exp(−x2) over [−∞,∞], real positive Gauss-Kronrod rules do not exist. Among the alternatives which are available in the literature, the anti-Gauss and average rules introduced by Laurie, and their modified versions, are of particular interest. In this paper, we investigate the properties of the modified anti-Gauss and average quadrature rules for ωα, and we determine the degree optimal average rules by proving that for each n-point Gauss rule for ωα there exists a unique average rule with the precise degree of exactness 2n+3. We also give some numerical examples to test the performance of the average rules obtained in this paper