# Asymptotic behaviour of the Christoffel functions on the Unit Ball in the presence of a Mass on the Sphere

## TL;DR

This paper studies the asymptotic behavior of Christoffel functions on the unit ball with a mass on the sphere, introducing new orthogonal polynomials and analyzing their properties.

## Contribution

It introduces a new family of multivariate orthogonal polynomials with a mass on the sphere and analyzes their asymptotic behavior, connecting them to classical polynomials and spherical harmonics.

## Key findings

- Connection formulas between new and classical polynomials
- Representation in terms of spherical harmonics
- Asymptotic analysis of Christoffel functions

## Abstract

We present a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes a mass uniformly distributed on the sphere. First, connection formulas relating these multivariate orthogonal polynomials and the classical ball polynomials are obtained. Then, using the representation formula for these polynomials in terms of spherical harmonics analytic properties will be deduced. Finally, we analyze the asymptotic behaviour of the Christoffel functions.