Approximation of functions by a new family of generalized Bernstein operators

The main object of this paper is to construct a new generalisation of the Bernstein operator, depending on a non-negative real parameter. We investigate some elementary properties of this operator, such as end point interpolation, linearity and positivity, etc. By using these generating operators, we provide another proof of the Weierstrass Approximation Theorem. We give the detailed proofs to the rate of convergence and Voronovskaja type asymptotic estimate formula for the operators. Moreover, an upper bound for the error is obtained in terms of the usual modulus of continuity. Shape preserving properties of the generalised Bernstein operators are also studied. It is proved that monotonic or convex functions produce monotonic or convex generalized Bernstein polynomials. (C) 2017 Elsevier Inc. All rights reserved.