Polynomial meshes on algebraic sets

TL;DR

This work develops a general framework to construct weakly admissible (and in many cases optimal) polynomial meshes on compact subsets of algebraic sets by pulling back norming sets from a projection to $\mathbb{C}^N$. A division-inequality approach ties the degree growth of polynomials on an algebraic set to that on a projection, enabling transfer of AM/WAM/OPM properties from a base set $K$ to lifted sets $E$ on hypersurfaces $V(s)$ and, in some cases, to higher codimension via iterative divisions. The paper provides precise degree bounds and dimension formulas, explicit index-shifting rules (e.g., $\ell(n)$) for key cases, and several concrete constructions with rigorous cardinality and constant estimates. Classical results are recovered for 1D domains, while nontrivial algebraic examples (sphere, spherical lune, Viviani’s window, and a cubic surface) illustrate the method’s reach and include numerical verifications (interpolation, least-squares, Lebesgue constants) in Matlab. The techniques offer practically computable, scalable norming sets for multivariate polynomial approximation on algebraic sets, with clear implications for interpolation accuracy and stability.

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.

Figures (8)

• Figure 1: The curves in red represent the boundary of the set $E$ in Example \ref{['Ex5']}. The black dots are the points of the disk $K$ that define the norming set $\mathcal{A} _7={\mathcal{M}}_{10,7}$, used to determine $\mathcal{B} _2$ on $E$. In particular, $\mathrm{card}\, \mathcal{A}_7=100$ and $\|p\|_K \leq 4.86 \|p\|_{\mathcal{A} _n}$, $p \in {\mathcal{P}}_7(K)$.
• Figure 2: The black dots are the points of the norming set $\mathcal{B} _2$ for $E$ in Example \ref{['Ex5']}. In particular, $\mathrm{card}\, \mathcal{B}_2=200$ and for any $n$ we have $\|p\|_E \leq 5.23 \|p\|_{\mathcal{B} _n}$, $p \in {\mathcal{P}}_n(E)$.
• Figure 3: The norming set $\mathcal{B} _4$ for Viviani's window $E$ in Example \ref{['VivWin']}, $\lambda=20$, $\mathrm{card}\, \mathcal{B}_4=80$ and $C(\mathcal{B}_4)=3.8$.
• Figure 4: The norming set $\mathcal{B} _4$ for Viviani's window $E$ in Example \ref{['VivWin']}, $\lambda=17$, $\mathrm{card}\, \mathcal{B}_4=68$ and $C(\mathcal{B}_4)=5.2$.
• Figure 5: From top to bottom, left to right, we plot the errors in approximating the functions $f_1$, $f_2$, $f_3$, $f_4$ by means of interpolation at Approximate Fekete Points (AFP), Discrete Leja Points (DLP) and least-squares on optimal Admissible meshes ${\mathcal{B}}_n$ for $n=1,2,\ldots,16$, on the set $E$ described in Example \ref{['Ex5']}.

Theorems & Definitions (31)

• Definition 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Definition 3.1