Renorming spaces with greedy bases

JOURNAL OF APPROXIMATION THEORY

卷:188

Article

Dilworth, S. J.¹  Kutzarova, D.^2,3  Schlumprecht, Th.^4,5  Zsak, A.⁶ 

[1] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

[2] Bulgarian Acad Sci, Inst Math, Sofia, Bulgaria

[3] Univ Illinois, Dept Math, Urbana, IL 61801 USA

[4] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

[5] Czech Tech Univ, Fac Elect Engn, Prague 16627, Czech Republic

[6] Univ Cambridge Peterhouse, Cambridge CB2 1RD, England

关键词: m-term approximation;    Greedy basis;    Democratic basis;    Renorming;    Fundamental function;   

DOI  :  10.1016/j.jat.2014.09.001

来源: Elsevier

【 摘 要 】

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given epsilon > 0, so that the basis becomes (1+epsilon)-democratic, and hence (2+epsilon)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1+ epsilon)greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in L-p [0, 1], 1 < p < infinity), and in dyadic Hardy space H-1, as well as the unit vector basis of Tsirelson space. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free