Random points are optimal for the approximation of Sobolev functions

09/23/2020

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by   David Krieg, et al.

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We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space W_p^s(Ω) on bounded convex domains Ω⊂ℝ^d in the L_q-norm if q<p. More generally, we characterize the quality of arbitrary sampling points P⊂Ω via the L_γ(Ω)-norm of the distance function dist(·,P), where γ=s(1/q-1/p)^-1 if q<p and γ=∞ if q≥ p. This improves upon previous characterizations based on the covering radius of P.