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
We consider the dilation operators in the frame of Besov spaces with 1 . If s > 0, is a bounded linear operator from 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 . 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 , where . It was observed already in  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 (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.