On a functional equation connected to Hermite quadrature rule

作     者：Koclega-Kulpa, Barbara Szostok, Tomasz 

作者机构：Silesian Univ Inst Math PL-40007 Katowice Poland 

出 版 物：《JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS》 (数学分析与应用杂志)

年 卷 期：2014年第414卷第2期

页      面：632-640页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Functional equations Approximate integration Polynomial functions Quadrature rules 

摘      要：In this paper we deal with the functional equation F(y) - F(x) = (y - x)[alpha f (x) + beta f (x + y/2) +alpha f(y)] + (y - x)(2) [g(y) - g(x), which is connected to Hermite quadrature rule. It is easy to note that particular cases of this equation generalize many well known functional equations connected to quadrature rules and mean value theorems. Thus the set of solutions is too complicated to be described completely and therefore we prove that (under some assumptions) all solutions of the above equation must be polynomials. We obtain the aforementioned result using a lemma proved by M. Sablik, however this lemma works only in case beta not equal 0. Taking beta = 0, we obtain the following equation F(y) - F(x) = (y - x)[f(x) + f (y)] + (y - x)(2) [g(y) - g (x)], which is also solved in the paper. (C) 2014 Elsevier Inc. All rights reserved.