Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa
Let \(\sigma\) be a positive measure on a compact interval \([a,\,b]\). A positive \(n\)-point quadrature formula has the form \[ \sum_{j=1}^n \lambda_{n,j}\,f(x_{n,j}) \approx \int_a^b f(x) \,d\sigma(x) \] with distinct abscissas \(x_{n,j}\in [a,\,b]\) and positive weights \(\lambda_{n,j}\). As known, there exist \(n\)-point Gauss rules with degree of precision \(2n-1\). Further, there exist \(n\)-point (left resp. right) Radau rules with abscissa \(a\) resp. \(b\) and with degree of precision \(2n-2\), and an \(n\)-point Lobatto rule with abscissas \(a\), \(b\) and with degree of precision \(2n-3\). In this paper, the authors show that for a given \(\theta\in (a,\,b)\), there exist an \(n\)-point quasi Gauss rule, \(n\)-point quasi (left/right) Radau rule, resp. \(n\)-point quasi Lobatto rule with the additional abscissa \(\theta\) and that this quadrature formula has the degree of precision \(2n-2\), \(2n-3\), resp. \(2n-4\). A central role in this study is played by quasi-orthogonal polynomials.
scientific article; zbMATH DE number 6418715