Rota–Baxter operators on the polynomial algebra, integration, and averaging operators

Shanghua Zheng, Li Guo and Markus Rosenkranz

Vol. 275 (2015), No. 2, 481–507

DOI: 10.2140/pjm.2015.275.481

Abstract

The concept of a Rota–Baxter operator is an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota–Baxter operators and integrals in the case of the polynomial algebra $k\left[x\right]$. We consider two classes of Rota–Baxter operators, monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial Rota–Baxter operators. For the second class, we make use of the double product on Rota–Baxter algebras.

Keywords

Rota–Baxter operator, averaging operator, integration, monomial linear operator

Mathematical Subject Classification 2010

Primary: 16W99

Secondary: 45N05, 47G10, 12H20

Milestones

Received: 30 May 2014

Accepted: 9 December 2014

Published: 15 May 2015