Shape-Preserving and Convergence Properties for the q-Szász-Mirakjan Operators for Fixed q∈(0,1)‎

Abstract EN

We introduce a q-generalization of Szász-Mirakjan operators Sn,q and discuss their properties for fixed q∈(0,1).

We show that the q-Szász-Mirakjan operators Sn,q have good shape-preserving properties.

For example, Sn,q are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity.

For fixed q∈(0,1), we prove that the sequence {Sn,qf} converges to B∞,q(f) uniformly on [0,1] for each f∈C[0, 1/(1-q)], where B∞,q is the limit q-Bernstein operator.

We obtain the estimates for the rate of convergence for {Sn,qf} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.