Shape-preserving properties of a new family of generalized Bernstein operators

Abstract

Abstract In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called (α,q) $(\alpha,q)$-Bernstein operators, denoted by Tn,q,α(f) $T_{n,q,\alpha}(f)$. We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of Tn,q,α(f) $T_{n,q,\alpha}(f)$ to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to f(x) $f(x)$. We also obtain the monotonicity with n and q of Tn,q,α(f) $T_{n,q,\alpha}(f)$.

Keywords

• Bernstein operators
• q-integers
• Shape-preserving
• Basis function
• Monotonicity