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On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases

Congpei An, Alvise Sommariva, Marco Vianello

TL;DR

This work analyzes the weak Marcinkiewicz-Zygmund constants for multivariate cubature rules and their influence on polynomial approximation via orthogonal bases. It characterizes $A$ and $B$ as the extremal eigenvalues of the Gram matrix $G=(S(\phi_i\phi_j))$ and uses $\eta=\max\{|1-A|,|1-B|\}$ to assess stability, also proving a negative result for Gaussian rules when $n$ crosses $\lfloor m/2\rfloor$. Through extensive numerical experiments on the interval, square, disk, triangle, and sphere, the authors compare hyperinterpolation, unfettered hyperinterpolation, and least-squares projections across a variety of rules (Gauss-Legendre, Clenshaw-Curtis, Padua, spherical designs, QMC Halton), identifying regimes where $\eta<1$ and conditioning remains tractable. The study highlights practical rule choices (e.g., Clenshaw-Curtis, Padua, certain spherical designs, sufficiently large QMC folds) that enable stable, accurate polynomial approximation with weak MZ constants, and provides open-source Matlab tools to reproduce and extend the results.

Abstract

We compute numerically the $L^2$ Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test some relevant rules on domains such as the interval, the square, the disk, the triangle, the cube and the sphere. The approximation power of the corresponding least squares (LS) projection is compared with standard hyperinterpolation and its recently proposed ``exactness-relaxed'' version. The Matlab codes used for these tests are available in open-source form.

On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases

TL;DR

This work analyzes the weak Marcinkiewicz-Zygmund constants for multivariate cubature rules and their influence on polynomial approximation via orthogonal bases. It characterizes and as the extremal eigenvalues of the Gram matrix and uses to assess stability, also proving a negative result for Gaussian rules when crosses . Through extensive numerical experiments on the interval, square, disk, triangle, and sphere, the authors compare hyperinterpolation, unfettered hyperinterpolation, and least-squares projections across a variety of rules (Gauss-Legendre, Clenshaw-Curtis, Padua, spherical designs, QMC Halton), identifying regimes where and conditioning remains tractable. The study highlights practical rule choices (e.g., Clenshaw-Curtis, Padua, certain spherical designs, sufficiently large QMC folds) that enable stable, accurate polynomial approximation with weak MZ constants, and provides open-source Matlab tools to reproduce and extend the results.

Abstract

We compute numerically the Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test some relevant rules on domains such as the interval, the square, the disk, the triangle, the cube and the sphere. The approximation power of the corresponding least squares (LS) projection is compared with standard hyperinterpolation and its recently proposed ``exactness-relaxed'' version. The Matlab codes used for these tests are available in open-source form.
Paper Structure (10 sections, 3 theorems, 40 equations, 15 figures)

This paper contains 10 sections, 3 theorems, 40 equations, 15 figures.

Table of Contents

  1. Introduction
  2. On the computation of weak Marcinkiewicz-Zygmund constants
  3. A negative result about Gaussian rules
  4. Numerical examples
  5. The interval
  6. The square
  7. The disk and the triangle
  8. The sphere
  9. QMC rules
  10. Polynomial approximation by orthogonal bases

Key Result

Proposition 1.1

Let $\mathcal{G}_nf$ be the weighted Least-Squares polynomial defined by (S) and (orth), where $f\in C(\Omega)$ and the cubature rule satisfies the MZ property (MZD)-(eta). Then the following bound holds which, in case the cubature rule is exact on the constants, can be refined to

Figures (15)

  • Figure 1: Contour lines of Marcinkiewicz-Zygmund constants $\eta$ in (\ref{['etabound']}) for Gauss-Legendre (left) and Clenshaw-Curtis (right) rules with $ADE=m$, where $m=1,2,\ldots,20$ and $n=0,\ldots,30$.
  • Figure 2: Conditioning in norm 2 of the Gramian based on Gauss-Legendre and Clenshaw-Curtis rules with $ADE=m$, where $m=1,2,\ldots,20$ and $n=0,\ldots,30$. Green dot: $\hbox{cond}_2(G) \in [1,10)$. Black dot: $\hbox{cond}_2(G)\in [10^7,+\infty)$.
  • Figure 3: Contour lines of Marcinkiewicz-Zygmund constants $\eta$ in (\ref{['etabound']}) some cubature rules on the unit-square. Tensor product Gauss-Legendre rule (top-left), Padua points based rule (top-right), near-minimal (bottom-left) and Morrow-Patterson-Xu (bottom-right) rules.
  • Figure 4: Conditioning in norm 2 of the Gramian based on tensor-product Gauss-Legendre rule, Padua points based rule, near minimal rule and and Morrow-Patterson-Xu rule with $ADE=m$, where $m=1,2,\ldots,20$ and $n=0,\ldots,30$. Green dot: $\hbox{cond}_2(G) \in [1,10)$. Yellow dot: $\hbox{cond}_2(G) \in [10,10^ 2)$. Magenta dot: $\hbox{cond}_2(G)\in [10^4,10^7)$. Black dot: $\hbox{cond}_2(G)\in [10^7,+\infty)$.
  • Figure 5: Contour lines of Marcinkiewicz-Zygmund constants $\eta$ in (\ref{['etabound']}) for a couple of cubature rules on the unit-disk.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Proposition 1.1
  • Remark 1.1
  • Proposition 2.1: Eigenvalue Characterization of Weak Marcinkiewicz--Zygmund Constants
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1
  • Remark 2.4
  • Remark 2.5