Approximation by α$$ \alpha $$-Bernstein–Schurer operators and shape preserving properties via q$$ q $$-analogue

Our work in this article is to construct the $\alpha$- Bernstein–Schurer operators which includes the $q$-integers. For these new operators, we discuss the shape preserving properties, namely, monotonicity and convexity. Next, we study the uniformly global approximation in terms of the Ditzian–Totik modulus of continuity and calculate the local direct estimate by Lipschitz-maximal functions. In the last Voronovskaja-type approximation theorems are also presented.     

Carleson and sampling measures on Bernstein spaces on Siegel CR manifolds

In this paper, we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are $L^p$-integrable with respect to a natural measure. For these spaces, we prove necessary and sufficient conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.