Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines
The paper deals with the problem of monotonicity for quadrature formulae of Gauss type. Moreover, comparison theorems for monosplines are studied. Let \(S_{m-1,k}\) be a subspace of spline functions of degree m-1 with k fixed knots. Suppose that \(K(S_{m-1,k})\) is the so-called convexity cone corresponding to \(S_{m-1,k}\). Let \(\{Q_ i\}\) be quadrature formulae of Gauss type corresponding to subspaces \(S_{m-1,k_ i}\). Then monotonicity theorems for the sequence \(\{Q_ i(f)\}\) are established where f is a function in \(K(S_{m-1,k_ i})\) for all i. For example, functions are considered where \(f^{(m)}\) has no sign changes. Moreover, inclusion theorems concerning the integral \(\int fd\alpha (t)\) are obtained. The approach is based on moment theory and the relationship between moment theory, quadrature formulae and one-sided \(L_ 1\)-approximation. Monosplines are associated with quadrature formulae of Gauss type. Comparison theorems for monosplines are given using also the relationship to moment theory. This result can be contrasted with the general improvement theorem for monosplines of Karlin.
scientific article; zbMATH DE number 3888290