Weighted least-squares approximation in expected $L^2$ norm
Matthieu Dolbeault  1@  , Albert Cohen  2  
1 : Laboratoire Jacques-Louis Lions
Sorbonne Université, Centre National de la Recherche Scientifique : UMR_7598, Université de Paris : UMR_7598
2 : LJLL
Sorbonne Université UPMC Paris VI

We investigate the problem of approximating a function u in L^2 with a linear space of functions of dimension n, using only evaluations of u at m chosen points, with m of the order of n. A first approach, based on weighted least-squares at i.i.d random points, provides a near-best approximation of u, but requires m of order n log(n). To reduce the sample size while preserving the quality of approximation, we need a result on sums of rank-one matrices, which answers to the Kadison-Singer conjecture.

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