Approximation processes by multidimensional Bernstein-type exponential polynomials on the hypercube

TL;DR

This paper introduces a new family of multidimensional Bernstein-type exponential polynomials on the hypercube, analyzing their approximation capabilities and convergence properties.

Contribution

It develops a novel class of exponential polynomials on the hypercube and establishes their uniform convergence and approximation order.

Findings

Proves uniform convergence of the new operators.

Provides quantitative estimates of approximation order.

Shows the operators fix exponential functions and their squares.

Abstract

In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube $[0, 1]^d$ and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its square. In particular, we prove uniform convergence, by means of two different approaches, as well as a quantitative estimate of the order of approximation in terms of the modulus of continuity of the approximated function.