A new family of extended Gauss quadratures with an interior interval constraint

摘要

Starting from two sequences {Ĝa,c,n} and {Ĝd,b,n} of ordinary Gauss quadrature formulae with an orthogonality measure dσ on the open intervals (a,c) and (d,b), respectively. We construct a new sequence {Ĝa,b,e(n)} of extended Gaussian quadrature formulae for dσ on (a,b), which is based on some preassigned points, the nodes of Ĝa,c,n, Ĝd,b,n and the e(n) zeros contained in (c,d) of a nonclassical orthogonal polynomial on [a,b] with respect to a linear functional. The principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three-term recurrence formulae are related. Thus, a new class of Gaussian quadratures, having some nodes contained in a given interior interval, can be computed directly by standard software for ordinary Gauss quadrature formulae.