Journal of Function Spaces

Journal of Function Spaces / 2008 / Article

Open Access

Volume 6 |Article ID 610196 | https://doi.org/10.1155/2008/610196

Jan Vybíral, "On dilation operators and sampling numbers", Journal of Function Spaces, vol. 6, Article ID 610196, 30 pages, 2008. https://doi.org/10.1155/2008/610196

On dilation operators and sampling numbers

Academic Editor: Hans Triebel
Received01 May 2007

Abstract

We consider the dilation operators Tk:ff(2k.) in the frame of Besov spaces Bpqs(d) with 1 p,q. If s > 0, Tk is a bounded linear operator from Bpqs(d) into itself and there are optimal bounds for its norm, see [4, 2.3.1]. We study the situation in the case s = 0, an open problem mentioned also in [4]. It turns out, that new effects based on Littlewood-Paley theory appear. In the second part of the paper, we apply these results to the study of the so-called sampling numbers of the embedding id:Bpq1s1(Ω)Bpq20(Ω), where Ω=(0,1)d. It was observed already in [13] that the estimates from above for the norm of the dilation operator have their immediate counterpart in the estimates from above for the sampling numbers. In this paper we show that even in the limiting case s2=0 (left open so far), this general method supplies optimal results.

Copyright © 2008 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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