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September 2015

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Abstract of a paper by Stephane Chretien and Sebastien Darses
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "An elementary approach to the problem of column selection in a rectangular matrix" by Stephane Chretien and Sebastien Darses. Abstract: The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis; see \cite{BourgainTzafriri:IJM87,Tropp:StudiaMath08,Vershynin:IJM01} for methods using random column selection. More constructive approaches have been proposed recently; see the recent contributions of \cite{SpielmanSrivastava:IJM12,Youssef:IMRN14}. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, i.e. a matrix whose singular values all lie in $[1-\epsilon,1+\epsilon]$. We provide individual lower and upper bounds for each singular value of the extracted matrix at the price of conceding only one log factor in the number of columns, when compared to the Restricted Invertibility Theorem of Bourgain and Tzafriri. Our method is fully constructive and the proof is short and elementary. Archive classification: math.FA math.SP Remarks: 5 pages Submitted from: stephane.chretien(a)npl.co.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.00748 or http://arXiv.org/abs/1509.00748

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Abstract of a paper by Victor Bible and Richard J. Smith
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Smooth and polyhedral approximation in Banach spaces" by Victor Bible and Richard J. Smith. Abstract: We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(\Gamma)$, where $\Gamma$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf. Archive classification: math.FA Mathematics Subject Classification: 46B03 46B20 Remarks: 12 pages Submitted from: victor.bible(a)ucdconnect.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.00369 or http://arXiv.org/abs/1509.00369

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