Weighted $\theta$-Incomplete Pluripotential Theory

TL;DR

This paper develops a new branch of pluripotential theory by integrating weighted and $ heta$-incomplete polynomial concepts, establishing extremal functions, and analyzing their asymptotic behavior.

Contribution

It introduces weighted $ heta$-incomplete pluripotential theory, defining extremal functions and connecting them with orthonormal polynomials and Bergman function asymptotics.

Findings

Established a Siciak-Zahariuta type equality for $ heta$-incomplete polynomials

Demonstrated extremal functions can be recovered via orthonormal polynomials

Proved strong asymptotics of Bergman functions in this context

Abstract

Weighted pluripotential theory is a rapidly developing area; and Callaghan \cite{Callaghan} recently introduced $\theta$-incomplete polynomials in \cd for $d>1$. In this paper we combine these two theories by defining weighted $\theta$-incomplete pluripotential theory. We define weighted $\theta$-incomplete extremal functions and obtain a Siciak-Zahariuta type equality in terms of $\theta$-incomplete polynomials. Finally we prove that the extremal functions can be recovered using orthonormal polynomials and we demonstrate a result on strong asymptotics of Bergman functions in the spirit of \cite{BermanCn}.