About the operator creating secondary polynomials

TL;DR

This paper explores the operator creating secondary polynomials linked to orthogonal systems, introducing the concept of a reducer function that enables extension and explicit formulation of the operator, with applications to classical orthogonal polynomials.

Contribution

It introduces the reducer function in the context of secondary polynomial operators, providing a method to reverse extensions and explicitly characterize the operator.

Findings

Defined the reducer function with a key role in secondary polynomial operators.

Established conditions for the extension and explicit formulation of the operator.

Presented numerical applications involving Fourier coefficients of the reducer.

Abstract

We are studying here the classical operator creating secondary polynomials associated with an orthogonal system for a continuous probability density function on a real interval. We know it is possible with the coupling of Stietjes Transforms to build an auxiliary measure also called secondary measure, making associated polynomials orthogonal. One of the consequences of this definition is the possibility to extend this operator to a function having interesting isometric characters. Under some hypotheses, there appears in the construction of the secondary measure a function having a privileged role which we will develop here: we will call it the reducer which will enable us among other kings to reverse the studied extension and to make its adjunct explicit. We will also illustrate it, in the case of classical orthogonal polynomials with a few results on the Fourier's coefficients of this reducer with different numerical applications.