Linear spectral transformations of Carath\'eodory functions

TL;DR

This paper investigates linear spectral transformations of Carathéodory functions, exploring their relation to moment functionals and measures, and characterizing those arising from rational modifications versus more general transformations.

Contribution

It characterizes linear spectral transformations of Carathéodory functions, including those not derived from rational modifications, and analyzes their impact on associated functionals and measures.

Findings

Rational modifications are a subset of linear spectral transformations.

Many linear spectral transformations are not rational modifications.

Examples reveal complex relationships between moment functionals.

Abstract

In this paper we present some recent results concerning linear spectral transformations of Carath\'eodory functions. More precisely, given two Carath\'eodory functions related by a linear spectral transformation, we study the relation between the corresponding moment functionals and, in the positive definite case, the relation between the measures. We will see that rational modifications of functionals are included in the linear spectral transformations. However, we will show that there exist a huge class of linear spectral transformations which are not given by rational modifications of functionals. Indeed, we will characterize those linear spectral transformation which come from a rational modification. In the general case we will discuss the relation between the functionals involved in a linear spectral transformation, which allows us to identify the difficulties to connect the related functionals. Actually, several examples will show how amazing can be the relationships between the moment functionals associated with a linear spectral transformation.