Polynomial meshes on algebraic sets

TL;DR

This paper introduces a general method for constructing polynomial meshes on algebraic sets, enabling efficient approximation and norm estimation, with demonstrated numerical applications in interpolation and least-squares approximation.

Contribution

It provides a novel construction of polynomial weakly admissible meshes on algebraic hypersurfaces and other algebraic sets, extending their applicability and optimality in approximation tasks.

Findings

Constructed polynomial meshes are optimal in some cases.

Numerical tests show effectiveness in interpolation and approximation.

Meshes facilitate estimation of supremum norms on algebraic sets.

Abstract

Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on compact subsets of arbitrary algebraic hypersurfaces in C^{N+1}. They are preimages by a projection of meshes on compacts in C^N. The meshes constructed in this way are optimal in some cases. Our method can be useful also for certain algebraic sets of codimension greater than one. To illustrate applications of the obtained theorems, we first give a few examples and finally report some numerical results. In particular, we present numerical tests (implemented in Matlab), concerning the use of such optimal polynomial meshes for interpolation and least-squares approximation, as well as for the evaluation of the corresponding Lebesgue constants.