Pad\'e approximation for a multivariate Markov transform

TL;DR

This paper applies Padé approximation techniques to analyze a multivariate Markov transform, extending spectral theory concepts and providing new characterizations and cubature formulas for specific measures.

Contribution

It introduces a novel analysis of multivariate Markov transforms using Padé approximation, including rationality characterization and a cubature formula for certain measures.

Findings

Rationality of the multivariate Markov transform characterized by Hankel determinants

Development of a cubature formula for a special class of measures

Extension of spectral theory concepts to multivariate settings

Abstract

Methods of Pad\'e approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors, and which is generalizing the well-known in Spectral theory Stieltjes transform (Markov function) of one-dimensional measure. The first main result is a characterization of the rationality of the Markov transform via Hankel determinants. The second main result is a cubature formula for a special class of measures.