Optimal pointwise sampling for L2 approximation

Given a function u?L2=L2(D,?), where ? is measure on set D, and linear subspace Vn?L2 of dimension n, we show that near-best approximation u in Vn can be computed from near-optimal budget Cn pointwise evaluations u, with C>1 universal constant. The sampling points are drawn according to some random distribution, the by weighted least-squares method, error assessed expected L2 norm. This result improves results [6], [8] which require sub-optimal logarithmic factor, thanks sparsification strategy introduced [17], [18]. As consequence, obtain for any compact class K?L2 number ?Cnrand(K)L2 randomized setting dominated Kolmogorov n-width dn(K)L2. While our shows existence such properties, discuss remaining issues concerning its generation computationally efficient algorithm.