<abstract xmlns="http://www.w3.org/1999/xhtml"><p> We present a method of solving functional equations of the type <disp-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-009-0045-2#0003_graphic1.png"></inline-graphic></disp-formula> where <italic>f, F: P → P</italic> are unknown functions acting on an integral domain  P and parameteres <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-009-0045-2#0003_graphic2.png"></inline-graphic></inline-formula> are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation <disp-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-009-0045-2#0003_graphic3.png"></inline-graphic></disp-formula>(1) for <italic>f, F:</italic> ℝ → ℝ is solved without any regularity assumptions. It is worth noting that (1) stems from a well-known quadrature rule used in numerical analysis. </p></abstract>

We present a method of solving functional equations of the type  where f, F: P → P are unknown functions acting on an integral domain  P and parameteres  are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation (1) for f, F: ℝ → ℝ is solved without any regularity assumptions. It is worth noting that (1) stems from a well-known quadrature rule used in numerical analysis.

Some functional equations characterizing polynomials

Institute of Mathematics Silesian University Bankowa 14 PL-40-007 Katowice, POLAND

Department of Mathematics and Computer Science University of Bielsko-Biała Willowa 2 PL-43-309 Bielsko-Biała, POLAND

Tatra Mountains Mathematical Publications

{"article-title":"Some functional equations characterizing polynomials"}

We present a method of solving functional equations of the type  where f, F: P → P are unknown functions acting on an integral domain  P and parameteres  are...

Some functional equations characterizing polynomials

Published Online: Nov 12, 2012

Page range: 27 - 40

DOI: https://doi.org/10.2478/v10127-009-0045-2

Keywords
functional equation on integral domain, quadrature rules, polynomial functions

This content is open access.

Some functional equations characterizing polynomials

Published Online: Nov 12, 2012

Page range: 27 - 40

DOI: https://doi.org/10.2478/v10127-009-0045-2

Keywordsfunctional equation on integral domain, quadrature rules, polynomial functions

This content is open access.

Keywords
functional equation on integral domain, quadrature rules, polynomial functions