# Multivariate approximation of functions on irregular domains by weighted least-squares methods

## TL;DR

This paper develops weighted least-squares algorithms for approximating functions on irregular domains, enabling stable, quasi-optimal estimations with computational efficiency even without explicit basis functions.

## Contribution

It introduces a method to construct stable weighted least-squares estimators on irregular domains using surrogate bases, extending previous work to more complex geometries.

## Key findings

- Stable estimators achieved with m ~ n log n function evaluations.
- Surrogate basis construction depends on Christoffel function of domain and space.
- Numerical experiments confirm theoretical stability and accuracy.

## Abstract

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented.