Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Dustin G. Mixon
by alspach＠math.okstate.edu 30 Mar '14

30 Mar '14

This is an announcement for the paper "Explicit matrices with the Restricted Isometry Property: Breaking the square-root bottleneck" by Dustin G. Mixon. Abstract: Matrices with the restricted isometry property (RIP) are of particular interest in compressed sensing. To date, the best known RIP matrices are constructed using random processes, while explicit constructions are notorious for performing at the "square-root bottleneck," i.e., they only accept sparsity levels on the order of the square root of the number of measurements. The only known explicit matrix which surpasses this bottleneck was constructed by Bourgain, Dilworth, Ford, Konyagin and Kutzarova. This chapter provides three contributions to further the groundbreaking work of Bourgain et al.: (i) we develop an intuition for their matrix construction and underlying proof techniques; (ii) we prove a generalized version of their main result; and (iii) we apply this more general result to maximize the extent to which their matrix construction surpasses the square-root bottleneck. Archive classification: math.FA cs.IT math.CO math.IT Remarks: Book chapter, submitted to Compressed Sensing and its Applications Submitted from: dustin.mixon(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.3427 or http://arXiv.org/abs/1403.3427

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Abstract of a paper by A. Elbour, N. Machrafi, and M. Moussa
by alspach＠math.okstate.edu 30 Mar '14

30 Mar '14

This is an announcement for the paper "Weak compactness of almost limited operators" by A. Elbour, N. Machrafi, and M. Moussa. Abstract: The paper is devoted to the relationship between almost limited operators and weakly compacts operators. We show that if $F$ is a $\sigma $-Dedekind complete Banach lattice then, every almost limited operator $T:E\rightarrow F $ is weakly compact if and only if $E$ is reflexive or the norm of $F$ is order continuous. Also, we show that if $E$ is a $\sigma $-Dedekind complete Banach lattice then the square of every positive almost limited operator $ T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order continuous. Archive classification: math.FA Mathematics Subject Classification: 46B42 (Primary) 46B50, 47B65 (Secondary) Remarks: 5 pages Submitted from: azizelbour(a)hotmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.3348 or http://arXiv.org/abs/1403.3348

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 30 Mar '14

30 Mar '14

This is an announcement for the paper "WORTH property, Garc\'{i}a-Falset coefficient and Opial property of infinite sums" by Jan-David Hardtke. Abstract: We prove some results concerning the WORTH property and the Garc\'{i}a-Falset coefficient of absolute sums of infinitely many Banach spaces. The Opial property/uniform Opial property of infinite $\ell^p$-sums is also studied and some properties analogous to the Opial property/uniform Opial property for Lebesgue-Bochner spaces $L^p(\mu,X)$ are discussed. Archive classification: math.FA Mathematics Subject Classification: 46B20 46E40 Remarks: 22 pages Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.2647 or http://arXiv.org/abs/1403.2647

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Abstract of a paper by Sheng Zhang
by alspach＠math.okstate.edu 30 Mar '14

30 Mar '14

This is an announcement for the paper "Coarse quotient mappings between metric spaces" by Sheng Zhang. Abstract: We give a definition of coarse quotient mapping and show that several results for uniform quotient mapping also hold in the coarse setting. In particular, we prove that any Banach space that is a coarse quotient of $L_p\equiv L_p[0,1]$, $1<p<\infty$, is isomorphic to a linear quotient of $L_p$. It is also proved that $\ell_q$ is not a coarse quotient of $\ell_p$ for $1<p<q<\infty$ using Rolewicz's property ($\beta$). Archive classification: math.FA math.MG Submitted from: z1986s(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.1934 or http://arXiv.org/abs/1403.1934

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Abstract of a paper by B. Bongiorno, U. B. Darju, and L. Di Piazza
by alspach＠math.okstate.edu 30 Mar '14

30 Mar '14

This is an announcement for the paper "Lineability of non-differentiable Pettis primitives" by B. Bongiorno, U. B. Darju, and L. Di Piazza. Abstract: Let X be an in?nite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly di?erentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly di?erentiable primitives, is lineable, i.e., there is an in?nite dimensional vector space whose nonzero vectors belong to ND. Archive classification: math.FA Mathematics Subject Classification: 46G10, 28B05 Submitted from: ubdarj01(a)louisville.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.1908 or http://arXiv.org/abs/1403.1908

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Abstract of a paper by Deping Ye
by alspach＠math.okstate.edu 10 Mar '14

10 Mar '14

This is an announcement for the paper "New Orlicz affine isoperimetric inequalities" by Deping Ye. Abstract: The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_{\phi}$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $L_{\phi}$ affine and geominimal surface areas reveal that these affine invariants are essentially the infimum/supremum of $V_{\phi}(K, L^\circ)$, the Orlicz $\phi$-mixed volume of $K$ and the polar body of $L$, where $L$ runs over all star bodies and all convex bodies, respectively, with volume of $L$ equal to the volume of the unit Euclidean ball $B_2^n$. Properties for the Orlicz $L_{\phi}$ affine and geominimal surface areas, such as, affine invariance and monotonicity, are proved. Related Orlicz affine isoperimetric inequalities are also established. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: deping.ye(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.1643 or http://arXiv.org/abs/1403.1643

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Abstract of a paper by Anna Kaminska, Karol Lesnik, and Yves Raynaud
by alspach＠math.okstate.edu 10 Mar '14

10 Mar '14

This is an announcement for the paper "Dual spaces to Orlicz - Lorentz spaces" by Anna Kaminska, Karol Lesnik, and Yves Raynaud. Abstract: For an Orlicz function $\varphi$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the K\"othe dual of an Orlicz-Lorentz function space $\Lambda_{\varphi,w}$ or a sequence space $\lambda_{\varphi,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\int\varphi_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $\varphi_*$ is the complementary function to $\varphi$. The second one is stated in terms of the modular $\int_I \varphi_*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\int\psi(f^*/|g|)|g|: g\prec w\}=\int_I \psi((f^*)^0/w)w$ valid for any measurable function $f$ and Orlicz function $\psi$. Analogous identity and dual representations are also presented for sequence spaces. Archive classification: math.FA Mathematics Subject Classification: 42B25, 46B10, 46E30 Remarks: 25 pages Submitted from: klesnik(a)vp.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.1505 or http://arXiv.org/abs/1403.1505

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Abstract of a paper by Ioannis Gasparis
by alspach＠math.okstate.edu 10 Mar '14

10 Mar '14

This is an announcement for the paper "A proof of Rosenthal's $\ell_1$ Theorem" by Ioannis Gasparis. Abstract: A proof is given of Rosenthal's $\ell_1$ theorem. Archive classification: math.FA Mathematics Subject Classification: 46B03 Remarks: 5 pages Submitted from: ioagaspa(a)math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.1163 or http://arXiv.org/abs/1403.1163

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Abstract of a paper by Claudia Correa and Daniel V. Tausk
by alspach＠math.okstate.edu 10 Mar '14

10 Mar '14

This is an announcement for the paper "On the $c_0$-extension property for compact lines" by Claudia Correa and Daniel V. Tausk. Abstract: We present a characterization of the continuous increasing surjections $\phi:K\to L$ between compact lines $K$ and $L$ for which the corresponding subalgebra $\phi^*C(L)$ has the $c_0$-extension property in $C(K)$. A natural question arising in connection with this characterization is shown to be independent of the axioms of ZFC. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46E15, 54F05 Remarks: 12 pages Submitted from: tausk(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.0605 or http://arXiv.org/abs/1403.0605

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Abstract of a paper by A. Elbour, N. Machrafi, and M. Moussa
by alspach＠math.okstate.edu 10 Mar '14

10 Mar '14

This is an announcement for the paper "On the class of weak almost limited operators" by A. Elbour, N. Machrafi, and M. Moussa. Abstract: We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices $E$, $F$ for which every positive weak almost limited operator $T:E\rightarrow F$ is almost limited (resp. almost Dunford-Pettis). As consequences, we will give some interesting results. Archive classification: math.FA Mathematics Subject Classification: 46B42 (Primary) 46B50, 47B65 (Secondary) Submitted from: azizelbour(a)hotmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.0136 or http://arXiv.org/abs/1403.0136

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