70 research outputs found
An asymptotic property of Schachermayer's space under renorming
A Banach space X with closed unit ball B is said to have property 2-beta,
repsectively 2-NUC if for every \ep > 0, there exists \delta > 0 such that for
every \ep-separated sequence (x_n) in the unit ball B, and every x in B, there
are distinct indices m and n such that ||x + x_m + x_n|| < 3(1 - \delta),
respectively, ||x_m + x_n|| < 2(1 - \delta). It is shown that a Banach space
constructed by Schachermayer has property 2-beta but cannot be renormed to have
property 2-NUC
Isomorphisms and strictly singular operators in mixed Tsirelson spaces
We study the family of isomorphisms and strictly singular operators in mixed
Tsirelson spaces and their modified versions setting. We show sequential
minimality of modified mixed Tsirelson spaces T_M[(\mc{S}_n,\theta_n)]
satisfying some regularity conditions and present results on existence of
strictly singular non-compact operators on subspaces of mixed Tsirelson spaces
defined by the families (\mc{A}_n)_n and (\mc{S}_n)_n.Comment: 29 pages, no figure
Explicit constructions of RIP matrices and related problems
We give a new explicit construction of matrices satisfying the
Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n
satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This
overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence,
which are used in all previous explicit constructions of RIP matrices. Key
ingredients in our proof are new estimates for sumsets in product sets and for
exponential sums with the products of sets possessing special additive
structure. We also give a construction of sets of n complex numbers whose k-th
moments are uniformly small for 1\le k\le N (Turan's power sum problem), which
improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le
(\log N)^{4+o(1)}. This latter construction produces elementary explicit
examples of n by N matrices that satisfy RIP and whose columns constitute a new
spherical code; for those problems the parameters closely match those of
existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log
N)^{5/2+o(1)}.Comment: v3. Minor correction
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