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

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,q(f)} 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,q(f)} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.