Convergence of <i>α</i>-Bernstein-Durrmeyer operators about a collection of measures

Abstract This work offers a generalization of the α -Bernstein-Durrmeyer operators <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="fraktur">D</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi mathvariant="bold">Z</m:mi> </m:mrow> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:math> $\mathfrak{D}^{\alpha}_{m,\mathbf{Z}}$ about a collection Z of arbitrary measures, with the goal of building a bridge between approximation, probability, and measure theory. It has been discussed how to achieve pointwise convergence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="fraktur">D</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi mathvariant="bold">Z</m:mi> </m:mrow> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:math> $\mathfrak{D}^{\alpha}_{m,\mathbf{Z}}$ and constraints on measure collection have been developed for these operators’ convergence. With certain values of the measures considered, the generalization yields a range of Bernstein-like operators with their α variation. In order to handle a wider variety of p.l.o. and investigate their convergence, the paper comprises numerous operators, making it a useful tool.