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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.
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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 $(oplus_{n=1}^infty ell_p^n)_{ell_1}$, with $1<ple infty$, and $(oplus_{n=1}^infty ell_p^n)_{c_0}$, with $1le p<infty$ do not have a greedy bases. We prove as well that the space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$ has a 1-greedy basis if and only if $1leq p=qle infty$.
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.
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 these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.
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 supports of orthonormal wavelets in our framework are discontinuous in the frequency domain, yet this irregularity constraint can be avoided in frames, even with redundancy factor less than 2. In this paper, we focus on the extension of orthonormal wavelets to biorthogonal wavelets and show that the same obstruction of regularity as in orthonormal schemes exists in biorthogonal schemes. In addition, we provide a numerical algorithm for biorthogonal wavelets construction where the dual wavelets can be optimized, though at the cost of deteriorating the primal wavelets due to the intrinsic irregularity of biorthogonal schemes.
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 functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
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