The existence and construction of rational Gauss-type quadrature rules

摘要

Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss–Radau (m=1) and Gauss–Lobatto (m=2) quadrature formulas that approximate F{f}. These are quadrature formulas with n positive weights and with the n-m remaining nodes real and distinct, so that the quadrature is exact in a (2n-m)-dimensional space of rational functions. Further, we also consider the case in which the functional is defined by a positive bounded Borel measure on an interval, for which it is required in addition that the nodes are all in the support of the measure.