On a new method for controlling exponential processes

TL;DR

This paper introduces and analyzes a Bernstein exponential operator, extending classical polynomial control tools to exponential polynomials, with implications for multivariate data representation.

Contribution

It proves key properties of the Bernstein exponential operator and explores special exponential polynomials for multivariate data representation.

Findings

Established fundamental properties of the Bernstein exponential operator

Identified special exponential polynomials for multivariate data

Extended classical polynomial control methods to exponential functions

Abstract

Unlike the classical polynomial case there has not been invented up to very recently a tool similar to the Bernstein-Bezier representation which would allow us to control the behavior of the exponential polynomials. The exponential analog to the classical Bernstein polynomials has been introduced in a recent authors' paper which appeared in Constructive Approximations, and this analog retains all basic properties of the classical Bernstein polynomials. The main purpose of the present paper is to contribute in this direction, by proving some important properties of the "Bernstein exponential operator" which has been introduced. We also fix our attention upon some special type of exponential polynomials which are particularly important for the further development of theory of representation of Multivariate data.