Rota-Baxter operators on the polynomial algebras, integration and averaging operators

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[x]. 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.