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

We consider the convergence of Gauss-type quadrature formulas for the integral ∫∞ 0 f(x)ω(x)dx, where ω 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 = { ∑q−1 k=−p akx 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∞, then convergence holds for all functions f for which fω 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.