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

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 $Vn \subset L^2(\Omega)$, 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 $Vn$ 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 $Vn$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $Vn$ 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.