Discrete Jackson-type Operators via a Boolean Sum Approach

Abstract

In the present paper we consider Jackson-type operators obtained via a Boolean sum, as studied by Cao, Gonska et al. We also present an alternative to their classical method of discretization using appropriate quadrature formulas. Our goal is to obtain for the discretized operators the same degree of approximation as Cao and Gonska (in particular, DeVore-Gopengauz inequalities), at the same time making sure that the operators discretized by means of our method will inherit more properties from the initial operators than by using quadrature formulas. As an example, we will consider convolution-type operators that preserve convexity up to a certain order, and we will show that the operators discretized with our technique also preserve monotonicity and convexity.