Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials
William E. Smith, Ian H. Sloan, A. H. Opie · Mathematics of Computation · 1983
The paper discusses both theoretical properties and practical implementation of product integration rules of the form \[ ∫ − ∞ ∞ k ( x ) f ( x ) d x ≈ ∑ i = 1 n w n i f ( x n i ) , \int _{ - \infty }^\infty {k(x)f(x)\,dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} } \] where f is continuous, k is absolutely integrable, the nodes { x n i } \{ {x_{ni}}\} are roots of the Hermite polynomials H n ( x ) {H_n}(x) , and the weights { w n i } \{ {w_{ni}}\} are chosen so that the rule is exact if f is any polynomial of degree > n > n . Convergence of the rule to the exact integral as n → ∞ n \to \infty is pr