A New Class of Bernstein Operators with Shifted Knots and Statistical Approximation

A New Class of Bernstein Operators

Abstract

This article introduces a generalized Bernstein operators with two shifted nodes. We first establish fundamental approximation tools, including estimates for moments and central moments. Using these, we prove a Korovkin-type convergence theorem adapted to the scaling behavior induced by the fractional integral. Further, we discuss convergence theorems and order of approximation in terms of first order modulus of smoothness. Next, we study pointwise approximation results in terms of Peetre's K-functional, second order modulus of smoothness, Lipschitz type space and $r^{th}$ order Lipschitz type maximal function. Lastly, weighted approximation results and statistical approximation theorems are proved.