Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

Abstract:

The paper discusses both theoretical properties and practical implementation of product integration rules of the form \[ \int _{ - \infty }^\infty {k(x)f(x) dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} } \] where f is continuous, k is absolutely integrable, the nodes $\{ {x_{ni}}\}$ are roots of the Hermite polynomials ${H_n}(x)$, and the weights $\{ {w_{ni}}\}$ are chosen so that the rule is exact if f is any polynomial of degree $< n$. Convergence of the rule to the exact integral as $n \to \infty$ is proved for a wide class of functions f and k (including singular or oscillatory functions k), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over $[0,\infty )$ are also discussed.