Modified Stieltjes polynomials and Gauss-Kronrod quadrature rules
Whereas \(n\)-point Gaussian quadrature formulas exist for all \(n \in \mathbb N\) and all nonnegative and integrable weight functions, it is well known that the corresponding Kronrod extensions exist only for a subset of this class of weights. In order to find a remedy for this deficiency, the authors develop a class of modified Kronrod quadrature formulas where, in addition to the usual \(n\) Gaussian nodes, \(r\) further nodes are prescribed, with \(r \leq n\). One then seeks the remaining \(n+1-r\) nodes and suitable weights such that the associated \((2n+1)\)-point quadrature formula has degree of exactness \(3n+1-r\). Clearly, the case \(r=0\) amounts to the standard Kronrod construction. The main goal of the paper is to exploit the additional freedom obtained when choosing \(r > 0\), with a particular emphasis on the cases \(r=1\) with the additional prescribed node being at one end point of the interval of integration, and \(r=2\) with both end points being prescribed. The main result for these cases is that, for a very large class of weight functions and sufficiently large \(n\), the associated modified Kronrod quadrature formulas exist and have the desirable properties that (a) all nodes are simple, (b) all nodes are inside the interval of integration, (c) the free nodes interlace with the prescribed nodes, and (d) all weights are positive. A detailed investigation of the associated nodal polynomials (i.e.\ the generalized Stieltjes polynomials) shows that their asymptotic behaviour is very similar to the behaviour of classical orthogonal polynomials. The work seems to have some relations to the results of \textit{S. Ehrich} [ISNM, Int. Ser. Numer. Math. 112, 67--76 (1993; Zbl 0799.65024)].
scientific article; zbMATH DE number 6830163
scientific article