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Orthogonal sequences and regularity of embeddings into finite-dimensional spaces
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de Moura, E. Pinto and Robinson, James C. (2010) Orthogonal sequences and regularity of embeddings into finite-dimensional spaces. Journal of Mathematical Analysis and Applications, Vol.368 (No.1). pp. 254-262. doi:10.1016/j.jmaa.2010.02.052 ISSN 0022-247X.
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Official URL: http://dx.doi.org/10.1016/j.jmaa.2010.02.052
Abstract
This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c(0) of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X X has finite Assouad dimension. (C) 2010 Elsevier Inc. All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Mathematical Analysis and Applications | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-247X | ||||
Official Date: | 1 August 2010 | ||||
Dates: |
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Volume: | Vol.368 | ||||
Number: | No.1 | ||||
Number of Pages: | 9 | ||||
Page Range: | pp. 254-262 | ||||
DOI: | 10.1016/j.jmaa.2010.02.052 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | CAPES, Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | EP/G007470/1 (EPSRC) |
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