Rota–Baxter algebra

In mathematics, a Rota–Baxter algebra is an algebra, usually over a field k, together with a particular k-linear map R which satisfies the weight-θ Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota,Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.

Let A be a k-algebra with a k-linear map R on A and let θ be a fixed parameter in k. We call A a Rota-Baxter k-algebra and R a Rota-Baxter operator of weight θ if the operator R satisfies the following Rota–Baxter relation of weight θ:

The operator R:= θ id − R also satisfies the Rota–Baxter relation of weight θ.

Integration by Parts

Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let ${\displaystyle C(R)}$ be the algebra of continuous functions from the real line to the real line. Let :${\displaystyle f(x)\in C(R)}$ be a continuous function. Define integration as the Rota–Baxter operator

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