Sampling of real multivariate polynomials and pluripotential theory

TL;DR

This paper investigates stable sampling of multivariate real polynomials on algebraic varieties, establishing density conditions linked to pluripotential theory that generalize classical results for tori and orthogonal polynomials.

Contribution

It introduces a general framework for sampling polynomials on algebraic varieties and connects sampling density to pluripotential equilibrium measures, extending classical univariate and torus results.

Findings

Sampling density must exceed the weighted equilibrium measure density.

Generalizes Landau's results for tori to algebraic varieties.

Shows zeros of orthogonal polynomials equidistribute with respect to equilibrium measures.

Abstract

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when $M$ is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of $M$, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity.