On the eigenstructure of the $(\alpha,q)$-Bernstein operator

Abstract

The eigenvalues and eigenvectors of $(\alpha,q)$-Bernstein operators are unknown and not studied in the literature. As the main result of this article, the eigenvalues and eigenvectors of $(\alpha,q)$-Bernstein operators are obtained. Moreover, we will give the asymptotic behaviour of these eigenvalues and eigenvectors for all $q>0.$ Some eigenvectors for various values of $\alpha$ and $q$ are depicted.