Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines
type: Item
instance of: scholarly article
title: Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines
zbMATH Open document ID: 0557.41023
DOI: 10.1007/BF01405566
author: D. Kharzeev
published in: Numerische Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132940
review text: 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.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 3888290
zbMATH Keywords: monotonicity for quadrature formulae
zbMATH Keywords: comparison theorems
zbMATH Keywords: monosplines
zbMATH Keywords: moment theory
zbMATH Keywords: one-sided \(L_ 1\)-approximation
MaRDI profile type: Publication
cites work: Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes
cites work: Error estimates for interpolatory quadrature formulae
cites work: Best quadrature formulas and splines
cites work: The fundamental theorem of algebra for monosplines satisfying boundary conditions
cites work: Dividierte Differenzen und Monotonie von Quadraturformeln
cites work: Moment Theory for Weak Chebyshev Systems with Applications to Monosplines, Quadrature Formulae and Best One-Sided $L^1 $-Approximation by Spline Functions with Fixed Knots
description: scientific article; zbMATH DE number 3888290