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

We consider the convergence of Gauss-type quadrature formulas for the integral R 1 0 f(x)!(x)dx where ! is a weight function on the half line [0; 1). The n-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials \Gammap;q\Gamma1 = f P q\Gamma1 k=\Gammap a k x k g where p = p(n) is a sequence of integers satisfying 0 p(n) 2n and q = q(n) = 2n \Gamma p(n). It is proved that under certain Carleman-type conditions for the weight and when p(n) or q(n) goes to 1, then convergence holds for all functions f for which f! is integrable on [0; 1). Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line. 1 Introduction In this paper, we consider the very classical and yet up to date problem of approximating a definite integral I ff (f) = Z b a f(x)dff(x) (1.1) where ff is a distribution function, i.e. a real valued, bounde..