On the convergence of quadrature formulas related to an infinite interval

The quadrature formulas we shall discuss in this paper belong to the same type as the well known formula given by Gauss.They can be characterized as follows.Let p(x) denote a function which does not assume negative values in a given interval (a, b) and is subject to the condition that the integrals Cm = I xmp(x)dx, or moments of the distribution determined by p(x), exist.Under such circumstances it is always possible to find » real numbers Xl < X2 < Xi < • • • < X» belonging to the interval (a, b) together with n corresponding constants Ai, At, • • • , An such that the formula (1) f P(x)f(x)dx = Aif(Xi) + Atf(x2) + ■■■+ Anf(Xn) holds true whenever f(x) is a polynomial whose degree does not exceed 2»-1, and this property completely determines the numbers xi} x2, • • • , xK as well as the corresponding coefficients A\, A2, • • • , An.It is important to notice that these coefficients are all positive.For any function f(x) which is not a polynomial of degree g 2«-1 the right member of (1) ceases to represent exactly the integral in the left member, and we must add the remainder 2?» in order to have an exact equality:(2) f P(x)f(x)dx = Aif(xi) + A2f(Xi) + ■■■+ Anf(Xn) + Rn. JaFor any function f(x) possessing a derivative of order 2» the following expression for the remainder Rn can be obtained:(3) Rn = p*kq 2»!I P(x)u2(x)dx, Ja