Approximation properties for modified $(p,q)$-Bernstein-Durrmeyer operators

Abstract

We introduce modified $(p,q)$-Bernstein-Durrmeyer operators. We discuss approximation properties for these operators based on Korovkin type approximation theorem and compute the order of convergence using usual modulus of continuity. We also study the local approximation property of the sequence of positive linear operators $D_{n,p,q}^{\ast}$ and compute the rate of convergence for the function $f$ belonging to the class ${\rm Lip}_M(\gamma)$.

Keywords

• $(p, q)$-integer
• $(p, q)$-Bernstein-Durrmeyer operator
• $q$-Bernstein-Durrmeyer operator
• modulus of continuity
• positive linear operator
• Korovkin type approximation theorem