A new family of extended Gauss quadratures with an interior interval constraint (Q5939861)

Let \([a,b]\) be a finite interval, \(\sigma\) a nonnegative measure on \([a,b]\) and \((c,d)\) a subinterval with \(a<c<d<b\). The authors are concerned with a class of quadrature formulae of Gauss type based on a system of nodes \(A\) of the form \(A= U\cup X\cup Y \cup Z\), where \(U\) consists of preassigned nodes outside \((a,b)\), \(X,Z\) each consist of the \(n\) nodes of the Gauss quadratures on the intervals \((a,c)\) and \((d,b)\), respectively and \(Y\) is a matrix of \(e(n)\) unknown nodes inside the interval \((c,d)\). The unknown nodes are obtained as zeros of nonclassical orthogonal polynomials with respect to a linear functional (if these are contained in \((c,d)\)). For numerical purposes the nodes and weights can be calculated (by standard software) via eigenvalues and eigenvectors of a symmetric tridiagonal matrix.