The Cauchy Exponential of Linear Functionals on the Linear Space of Polynomials

Abstract

In this paper, we introduce the notion of the Cauchy exponential of a linear functional on the linear space of polynomials in one variable with real or complex coefficients using a functional equation by using the so-called moment equation. It seems that this notion hides several properties and results. Our purpose is to explore some of these properties and to compute the Cauchy exponential of some special linear functionals. Finally, a new characterization of the positive-definiteness of a linear functional is given.

Keywords

• cauchy power of linear functional
• cauchy exponential of linear functional
• weakly-regular linear functional
• regular linear functional
• positive-definite linear functional
• orthogonal polynomial sequence