On a functional equation connected to Hermite quadrature rule

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:414

Article

Koclega-Kulpa, Barbara¹  Szostok, Tomasz¹ 

[1] Silesian Univ, Inst Math, PL-40007 Katowice, Poland

关键词: Functional equations;    Approximate integration;    Polynomial functions;    Quadrature rules;   

DOI  :  10.1016/j.jmaa.2014.01.002

来源: Elsevier

【 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.

【 授权许可】

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