Metric entropy, n-widths, and sampling of functions on manifolds

We first investigate on the asymptotics of the Kolmogorov metric entropy and nonlinear n-widths of approximation spaces on some function classes on manifolds and quasi-metric measure spaces. Secondly, we develop constructive algorithms to represent those functions within a prescribed accuracy. The constructions can be based on either spectral information or scattered samples of the target function. Our algorithmic scheme is asymptotically optimal in the sense of nonlinear n-widths and asymptotically optimal up to a logarithmic factor with respect to the metric entropy

09391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems

From 20.09.09 to 25.09.09, the Dagstuhl Seminar 09391 Algorithms and Complexity for Continuous Problems was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Sampling recovery in uniform and other norms

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 $\sqrt{n}$ 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.Comment: change of title and substantial revision compared to v