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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 $vare>0$, so that the basis becomes $(1+vare)$-democratic, and hence $(2+vare)$-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 $(1+vare)$-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 $L_p[0,1]$, $1<p<infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.
We prove thatthe Banach space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1leq p leqinfty$ and $1<q<infty$. Furthermore, the Banach spaces $(oplus_{n=1}^infty ell_p^n)_{ell_1}$, wit
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
We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that
We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work, We show that
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for