# Polynomials associated to non-convex bodies

@inproceedings{Levenberg2021PolynomialsAT, title={Polynomials associated to non-convex bodies}, author={Norman Levenberg and Franck Wielonsky}, year={2021} }

Polynomial spaces associated to a convex body C in (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 VC,K for K ⊂ Cd compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.

## References

SHOWING 1-10 OF 12 REFERENCES

### Zero distribution of random sparse polynomials

- Mathematics
- 2015

We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope $P$ as their degree grow. We consider a large class of…

### Multivariate polynomial approximation in the hypercube

- Mathematics
- 2016

A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion…

### Random Polynomials and Pluripotential-Theoretic Extremal Functions

- Mathematics
- 2013

There is a natural pluripotential-theoretic extremal function VK,Q associated to a closed subset K of ℂm$\mathbb {C}^{m}$ and a real-valued, continuous function Q on K. We define random polynomials…

### Bernstein–Walsh Theory Associated to Convex Bodies and Applications to Multivariate Approximation Theory

- Mathematics
- 2017

We prove a version of the Bernstein–Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full…

### Pluripotential Theory and Convex Bodies: A Siciak–Zaharjuta Theorem

- MathematicsComputational Methods and Function Theory
- 2020

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. We define the {\it logarithmic indicator function} on ${\bf…

### Pluripotential theory and convex bodies: large deviation principle

- MathematicsArkiv för Matematik
- 2019

We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large…

### ZEROS OF RANDOM POLYNOMIALS ON C

- Mathematics
- 2007

For a regular compact set K in Cm and a measure μ on K satisfying the Bernstein-Markov inequality, we consider the ensemble PN of polynomials of degree N , endowed with the Gaussian probability…

### Pluripotential theory and convex bodies

- Mathematics
- 2016

A seminal paper by Berman and Boucksom exploited ideas from complex geometry to analyze the asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles over compact,…

### Bernstein-Markov: a survey

- Mathematics
- 2015

We give a survey of recent results, due mainly to the authors, concerning Bernstein-Markov type inequalities and connections with potential theory.

### Equidistribution of zeros of random holomorphic sections

- Mathematics
- 2013

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of…