A representation for the integral kernel of the composition of multivariate Bernstein-Durrmeyer operators

This paper presents a representation for the kernel of the composition of multivariate Bernstein-Durrmeyer operators for functions defined on the standard simplex in $\mathbb{R}^d$.

💡 Research Summary

The paper investigates the integral kernel that arises when two multivariate Bernstein–Durrmeyer operators are composed. The authors first recall the one‑dimensional situation, where the kernel of the composition of two classical Bernstein–Durrmeyer operators can be expressed either in terms of Legendre polynomials or, more compactly, as a sum of products of Bernstein basis polynomials with matching indices. This latter representation reveals that the composition can be written as a linear combination of the original operators and makes commutativity transparent.

Moving to the multivariate setting, the authors work on the standard simplex (S_d\subset\mathbb{R}^d) and use barycentric coordinates together with standard multi‑index notation. For a multi‑index (\alpha) they define the multivariate Bernstein basis polynomial
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