Sampling recovery in uniform and other norms
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We study the recovery of functions in the uniform norm based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best L_2-approximation from a given nested sequence of subspaces combined with bounds on the the Christoffel function of these subspaces. Besides an explicit bound, we obtain that linear algorithms using n samples are optimal up to a factor √(n) among all algorithms using arbitrary linear information. Moreover, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces. We also discuss results for approximation in more general seminorms, including L_p-approximation.
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