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

Abstract

We introduce a $q$-generalization of Szász-Mirakjan operators $S_{n,q}$ and discuss their properties for fixed $q\in (\mathrm{0,1})$. We show that the $q$-Szász-Mirakjan operators $S_{n,q}$ have good shape-preserving properties. For example, $S_{n,q}$ are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed $q\in (\mathrm{0,1})$, we prove that the sequence $\{S_{n,q}\left(f\right)\}$ converges to $B_{\infty ,q}(f)$ uniformly on $[\mathrm{0,1}]$ for each $f\in C[0, 1/(1-q)]$, where $B_{\infty ,q}$ is the limit $q$-Bernstein operator. We obtain the estimates for the rate of convergence for $\{S_{n,q}\left(f\right)\}$ by the modulus of continuity of $f$, and the estimates are sharp in the sense of order for Lipschitz continuous functions.