Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions

Abstract

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions wα(x)=xαe−x, α>−1, on (0,+∞) by the oscillatory factor exp(iζx), where ζ>0. For calculation of such integrals for analytic real-valued functions in D={z∈ℂ∣Rez≥0,Imz≥0} we construct polynomials orthogonal with respect to the linear functional L:P→ℂ, given by L[p]=∫0+∞p(x)wα(x)exp(iζx)dx, where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals.