An introduction to multiple orthogonal polynomials and Hermite-Pad\'e approximation

TL;DR

This paper introduces multiple orthogonal polynomials and Hermite-Padé approximation, focusing on Nikishin systems, their properties, and asymptotic behaviors, providing foundational knowledge for further research in approximation theory.

Contribution

It offers a concise overview of the theory of multiple orthogonal polynomials related to Nikishin systems, highlighting key properties and asymptotic behaviors.

Findings

Properties of zeros and their distribution

Asymptotic behaviors of multiple orthogonal polynomials

Relevance to Hermite-Padé approximation

Abstract

We present a brief introduction to the theory of multiple orthogonal polynomials on the basis of known results for an important class of measures known as Nikishin systems. For type I and type II multiple orthogonal polynomials with respect to such systems of measures, we describe some of their most relevant properties regarding location and distribution of zeros as well as their weak and ratio asymptotic behavior.