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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 $Ω\subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(Ω)$, 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 $L^2(Ω)$-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 $Ω$ is an irregular domain such that the analytic form of an $L^2(Ω)$-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 $Ω$ and $V_n$. Numerical results validating our analysis are presented.