Multivariate approximation of functions on irregular domains by weighted least-squares methods
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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⊂R^d. Given any n-dimensional approximation space V_n ⊂ L^2(Ω), the analysis in [4] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations m of the order n 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 [4]. If the basis also has product form, then the algorithms in [4] 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 in not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from V_n, again with m of the order n 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.
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