Liebe Mitarbeiter des INS, im Rahmen unseres Forschungsseminars "Mathematics of Computation" wird Glen Byrenheid (Bonn) zum Thema "Optimal sampling of d-variate periodic functions with respect to energy sparse grids" vortragen. Kurzzusammenfassung: Motivated by Yserentant's results for the regularity of eigenfunctions of the electronic Schrödinger operator it seems useful to study the approximation behavior of functions belonging to a space of dominating mixed smoothness. We are interested in finding good approximations of functions with this kind of regularity from finite dimensional subspaces, where the error is measured in the so-called energy-norm H^1. We propose subspaces (energy-norm sparse grids) and corresponding linear mappings which yield the optimal asymptotic error rate (approximation numbers) in terms of the number of degrees of freedom for periodic d-variate functions with mixed smoothness s>1. Moreover, the mentioned linear mappings only consider function evaluations of the function f to be approximated. In particular, we obtain sharp bounds for sampling numbers in this context, which coincide with the corresponding estimates of the approximation numbers of the respective embedding. The rate is poynomial of order s-1 without logarithmic perturbance which would occur for classical sparse grid methods. Our proofs rely on classical Bernstein and discrete Hardy type inequalities to obtain dyadic sampling representations, i.e., equivalent norms in the source space where only function values are questioned. The results are part of a joint work with W. Sickel (Jena), Dinh Dung (Hanoi) and Tino Ullrich (Bonn). Der Vortrag findet am kommenden Montag, den 27.1. um 14:15 Uhr im Raum 6.020 WE6 statt. Mit den besten Grüßen Tino Ullrich und Daniel Peterseim _______________________________________________ Ins-mitarbeiter mailing list Ins-mitarbeiter@ins.uni-bonn.de https://mail.ins.uni-bonn.de/mailman/listinfo/ins-mitarbeiter