Polynomials associated to non-convex bodies

TL;DR

This paper extends the theory of polynomial spaces to non-convex bodies, developing pluripotential theory and approximation results that generalize existing convex-body frameworks.

Contribution

It introduces pluripotential theory for polynomial spaces associated with non-convex bodies, expanding the scope of approximation and asymptotic analysis.

Findings

Development of $C$-extremal plurisubharmonic functions

Bernstein-Walsh type approximation results in non-convex setting

Asymptotic behavior of random polynomials

Abstract

Polynomial spaces associated to a convex body $C$ in $({\bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for $K\subset {\bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.