Approximation Numbers of Embeddings of Anisotropic Sobolev Spaces of Dominating Mixed Smoothness
Abstract

We investigate the approximation of d-variate periodic functions in anisotropic Sobolev spaces of dominating mixed (fractional) smoothness s̅ on the d-dimensional torus, where the approximation error is measured in the L2-norm. 

As it is well-known, in high dimensions functions from isotropic Sobolev spaces HS (Td) can not be approximated sufficiently fast (in the sense of approximation numbers of corresponding embeddings). One needs to switch to smaller spaces. Since the begin-ning of the sixties it is known that Sobolev spaces of dominating mixed smoothness(Td) may help. However, for very large dimensions even these classes are over-sized. A way out is to sort the variables in dependence of there importance (in our ease in dependence of the smoothness). We associate to each variable different smoothness assumptions. As smoother the function is with respect to the variable xl as weaker is the influence of this variable. This philosophy is reflected in the choice of the function space 111x(Td) characterized by the norm []. We assumes[] and [] for some number 1 < v < d. It will be the main aim of my talk to describe the behaviour of the approximation numbers [] in dependence of n, s̅, v and d. Almost all of our results will be based on an elementary lemma, simplyfying in this way also our earlier results with respect to this topic. This is joined work with Thomas Kühn (Leipzig) and Tina Ullrich (Bonn).
 

Speaker: Professor Winfried Sickel
Date: 3 July 2019
Time: 16:00pm - 17:00pm
PosterClick here

Biography

Institute of Mathematics, Friedrich-Schiller-University Jena