On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

Abstract

We consider the convergence of Gauss-type quadrature formulas for the integral f 0 ∞ f(x)ω(x)dx, where w is a weight function on the half line [0, ∞). The n-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials Λ -p,q-1 = {Σ k=-p q-1 p a k x k }, where p = p(n) is a sequence of integers satisfying 0 < p(n) < 2n and q = q(n) = 2n - p(n). It is proved that under certain Carleman-type conditions for the weight and when p(n) or q(n) goes to oo, then convergence holds for all functions f for which fw is integrable on [0, ∞). Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.