Durrmeyer type operators linked with Boas-Buck type polynomials
Naokant Deo, Sandeep Kumar
TL;DR
The paper introduces Baskakov-Durrmeyer type operators $\mathcal{B}_n$ and their weighted variant $\tilde{\mathcal{B}}_n$ tied to Boas-Buck polynomials via analytic generating functions, and investigates their moment structure and convergence properties. It establishes explicit moment and central-moment formulas, provides Korovkin-type uniform convergence criteria under natural limits of the generating functions, and develops convergence rates in Lipschitz-type and weighted spaces using the Ditzian-Totik modulus of smoothness and related functionals. A comprehensive rate-of-convergence analysis is given for functions of bounded variation, with decompositions and bounds that reveal the dependence on the variation of derivatives and growth parameters. The work also extends the operators to a Szász-type Durrmeyer variant $\mathcal{B}_n^{*}$, demonstrating that the same convergence framework and rate results apply, thereby broadening the applicability of Boas-Buck polynomial-based approximation operators.
Abstract
The present article intends to introduce the sequence of Baskakov-Durrmeyer type operators linked with the generating functions of Boas-Buck type polynomials. After calculating the moments, including the limiting case of central moments of the constructed sequence of operators, in the subsequent sections, we estimate the convergence rate using the Ditzian-Totik modulus of smoothness and some convergence results in Lipchitz-type space.