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تسجيل مستخدم جديدOptimal pointwise sampling for $L^2$ approximation
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Given a function $uin L^2=L^2(D,mu)$, where $Dsubset mathbb R^d$ and $mu$ is a measure on $D$, and a linear subspace $V_nsubset L^2$ of dimension $n$, we show that near-best approximation of $u$ in $V_n$ can be computed from a near-optimal budget of $Cn$ pointwise evaluations of $u$, with $C>1$ a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected $L^2$ norm. This result improves on the results in [6,8] which require a sampling budget that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class $mathcal Ksubset L^2$ that the sampling number $rho_{Cn}^{rm rand}(mathcal K)_{L^2}$ in the randomized setting is dominated by the Kolmogorov $n$-width $d_n(mathcal K)_{L^2}$. While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.
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