Local saturation of conservative operators
type: Item
instance of: scholarly article
title: Local saturation of conservative operators
zbMATH Open document ID: 1058.41017
DOI: 10.1023/A:1024656217794
published in: Acta Mathematica Hungarica
publication date: 2004-09-07T00:00:00Z
review text: The authors state a result about the local saturation of sequences of linear operators that preserve the sign of \(k\)-th derivative of the functions. Let \(I\subset \mathbb R\), \(i\in \mathbb N_0=\mathbb N\cup \{0\}\). Let \(C^i(I)\) be the space of all real-valued and \(i\)-times continuously differentiable functions defined on \(I\) and \(D^i\) be the \(i\)-th differential operator with \(C(I)=C^0(I)\) the space of all continuous functions defined on \(I\) and \(D^0\) the identity operator, respectively. Besides, \(C^i_B(I)\) denotes the subspace formed by the functions of \(C^i(I)\) which are bounded on \(I\). A function \(f\in C^i(I)\) is said to be \(i\)-convex if \(D^i f\geq 0\) on \(I\) and a linear operator is said to be \(i\)-convex if it maps \(i\)-convex functions onto \(i\)-convex functions. Definition. Let \(L(y)=0\) denote the second-order linear differential equation \(L(y)\equiv D^2 y+p_1(t)D^1 y+p_2(t)y=0\). Consider an interval \((a,b)\) such that \(L(y)=0\) has a unique solution, continuous in \((a,b)\), taking any given real values \(y_1\) and \(y_2\) at any two given \(t_1\) and \(t_2\) within \((a,b)\). Assume that \(p_1(t)\) and \(p_2(t)\) are continuous and differentiable whenever this is required. A real function \(f(t)\) defined in \((a,b)\) is said to be \(\text{sub}\,-(L)\) in \((a,b)\) if \(f(t)\leq S(f,t_1,t_2)(t)\) for every \(t\), \(t_1\), \(t_2\) such that \(a<t_1<t<t_2<b\), \(S(f,t_1,t_2)\) being the solution of \(L(y)=0\) taking the values \(f(t_1)\) and \(f(t_2)\) at \(t_1\) and \(t_2\). Further, let \(I\) be a closed real interval, \(k\in \mathbb N_0\) and \(L_n:C^k(I)\to C^k(I)\) satisfying the following shape preserving property and asymptotic formula: (H1) for all \(n\in \mathbb N\), \(L_n\) is \(k\)-convex, (H2) there exist a sequence \(\lambda_n\) of real positive numbers, and a function \(p\in C^k(I)\) strictly positive on \(\text{Int}\,(I)\) such that for all \(g\in C^k_B(I)\), \(k+2\)-times differentiable in some neighb…
reviewed by: Zoltán Finta
Mathematics Subject Classification ID: 41A40
zbMATH DE Number: 2098983
zbMATH Keywords: saturation
zbMATH Keywords: \(k\)-convex operators
author: Daniel Cárdenas Morales
author: Pedro Garrancho
MaRDI profile type: Publication
full work available at URL: a:1024656217794
OpenAlex ID: W1833206538
description: scientific article; zbMATH DE number 2098983