Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898499 | Journal of Complexity | 2018 | 34 Pages |
Abstract
We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed-(quasi-)norm embeddings âpb(âqd)âªârb(âud) given that pâ¤r and qâ¤u, with emphasis on cases with pâ¤1 and/or qâ¤1. These cases turn out to be related to structured sparsity. We obtain sharp bounds in a number of interesting parameter constellations. Our new matching bounds for the Gelfand numbers of the embeddings of â1b(â2d) and â2b(â1d) into â2b(â2d) imply optimality assertions for the recovery of block-sparse and sparse-in-levels vectors, respectively. In addition, we apply our sharp estimates for âpb(âqd)-spaces to obtain new two-sided estimates for the Gelfand numbers of multivariate Besov space embeddings in regimes of small mixed smoothness. It turns out that in some particular cases these estimates show the same asymptotic behavior as in the univariate situation. In the remaining cases they differ at most by a loglog factor from the univariate bound.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Sjoerd Dirksen, Tino Ullrich,