A new quadrature scheme based on an Extended Lagrange Interpolation process

Let w(x) = e(-x beta) x(alpha), (w) over bar (x) = xw(x) and let {p(m)(w)}(m), {p(m)((w) over bar)}(m) be the corresponding sequences of orthonormal polynomials. Since the zeros of p(m+1) (w) interlace those of p(m)((w) over bar), it makes sense to construct an interpolation process essentially based on the zeros of Q(2m+1) := Pm+1 (w)P-m((w) over bar), which is called "Extended Lagrange Interpolation". In this paper the convergence of this interpolation process is studied in suitable weighted L-1 spaces, in a general framework which completes the results given by the same authors in weighted L-u(p) ((0, +infinity)), 1 <= p <= infinity (see [31], [28]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type integral(+infinity)(0) f(x)k(x, y)u(x)dx, u(x) = e(-x beta) x(gamma) (1+x)(lambda), gamma > -1, lambda is an element of R+, where the kernel k(x, y) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23]. It is shown that the "compound quadrature sequence" represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence. (C) 2017 IMACS. Published by Elsevier B.V. All rights reserved.