A representation for the integral kernel of the composition of multivariate Bernstein-Durrmeyer operators
Ulrich Abel, Ana Maria Acu, Margareta Heilmann, Ioan Rasa
TL;DR
The paper tackles kernel representations for the composition of multivariate Bernstein-Durrmeyer operators on the standard simplex. It generalizes previous univariate/multivariate results for the kernels of two and three operator compositions to the case of M_m ∘ M_n and beyond. The main contribution is an explicit kernel representation that involves diagonal Bernstein terms B_ell(x) B_ell(y) and a combinatorial prefactor, enabling a linear combination decomposition of the composition. The authors derive the result from Bernstein basis properties, inner product identities, and combinatorial manipulations, highlighting potential commutativity and iterative extensions. Future work will extend these ideas to higher-order iterates M_n^r and more general multivariate settings.
Abstract
This paper presents a representation for the kernel of the composition of multivariate Bernstein-Durrmeyer operators for functions defined on the standard simplex in $\mathbb{R}^d$.