Constructing measures with identical moments

TL;DR

This paper constructs a set of measures with identical moments without relying on orthogonal polynomials and develops a Nevanlinna-type parametrization to understand their analytic structure.

Contribution

It introduces a novel construction of measures with identical moments independent of orthogonal polynomial theory and establishes a new Nevanlinna-type parametrization for these measures.

Findings

Constructed measures all share identical moments.

Developed a Nevanlinna-type parametrization for these measures.

Revealed the analytic structure behind the Nevanlinna parametrization.

Abstract

The Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization.