Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Matthew Daws, Richard Haydon, Thomas Schlumprecht, and Stuart White
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Shift invariant preduals of $\ell_1(\Z)$" by Matthew Daws, Richard Haydon, Thomas Schlumprecht, and Stuart White. Abstract: The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is equivalent to making the natural convolution multiplication on $\ell_1(\Z)$ separately weak$*$-continuous and so turning $\ell_1(\Z)$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_0(K)$ (for countable locally compact $K$) and $\ell_1(\Z)$ is $c_0(\Z)$. We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak$^*$-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to $c_0$. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of $\Z$. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to $c_0$. Archive classification: math.FA Remarks: 31 pages Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5696 or http://arXiv.org/abs/1101.5696

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Abstract of a paper by Maxim V. Balashov and Dusan Repovs
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order" by Maxim V. Balashov and Dusan Repovs. Abstract: We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius. Archive classification: math.FA math.GN Mathematics Subject Classification: 46C05, 54C60, 46N10, 32F17 Citation: J. Math. Anal. Appl. 377:2 (2011), 754-761 Submitted from: dusan.repovs(a)guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5685 or http://arXiv.org/abs/1101.5685

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Abstract of a paper by Cleon S. Barroso, Ondrej F.K. Kalenda and Pei-Kee Lin
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "On the approximate fixed point property in abstract spaces" by Cleon S. Barroso, Ondrej F.K. Kalenda and Pei-Kee Lin. Abstract: Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual and $Z$ a subset of $X^*$. In this paper, we establish some results concerning the $\sigma(X,Z)$-approximate fixed point property for bounded, closed convex subsets $C$ of $X$. Three major situations are studied. First when $Z$ is separable in the strong topology. Second when $X$ is a metrizable locally convex space and $Z=X^*$, and third when $X$ is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fr\'echet-Urysohn property for certain sets with regarding the $\sigma(X,Z)$-topology. The support tools include the Brouwer's fixed point theorem and an analogous version of the classical Rosenthal's $\ell_1$-theorem for $\ell_1$-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces. Archive classification: math.FA math.GN Mathematics Subject Classification: 47H10, 46A03 Remarks: 14 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5274 or http://arXiv.org/abs/1101.5274

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Abstract of a paper by A. Thiago L. Bernardino
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Remarks on cotype absolutely summing multilinear operators" by A. Thiago L. Bernardino. Abstract: In this short note we present some new results concerning cotype and absolutely summing multilinear operators. Archive classification: math.FA Remarks: 5 pages Submitted from: thiagodcea(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5119 or http://arXiv.org/abs/1101.5119

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Abstract of a paper by Boris Rubin
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Elementary inversion of Riesz potentials and Radon-John transforms" by Boris Rubin. Abstract: New simple proofs are given to some elementary approximate and explicit inversion formulas for Riesz potentials. The results are applied to reconstruction of functions from their integrals over Euclidean planes in integral geometry. Archive classification: math.FA Remarks: 9 pages Submitted from: borisr(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.5105 or http://arXiv.org/abs/1101.5105

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Abstract of a paper by Daniel Fresen
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "A multivariate Gnedenko law of large numbers" by Daniel Fresen. Abstract: We show that the convex hull of a large i.i.d. sample from a non-vanishing log-concave distribution approximates a pre-determined body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For p-log-concave distributions with p>1 (such as the normal distribution where p=2) we also have approximation in the Hausdorff distance. These are multivariate versions of the Gnedenko law of large numbers which gaurantees concentration of the maximum and minimum in the one dimensional case. We give three different deterministic bodies that serve as approximants to the random body. The first is the floating body that serves as a multivariate quantile, the second body is given as a contour of the density function, and the third body is given in terms of the Radon transform. We end the paper by constructing a probability measure with an interesting universality property. Archive classification: math.PR math.FA Mathematics Subject Classification: 60D05, 60F99, 52A20, 52A22, 52B11 Remarks: 18 pages Submitted from: djfb6b(a)mail.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4887 or http://arXiv.org/abs/1101.4887

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Abstract of a paper by Guohui Song, Haizhang Zhang
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Reproducing kernel Banach spaces with the l1 norm II: Error analysis for regularized least square regression" by Guohui Song, Haizhang Zhang. Abstract: A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning. Archive classification: stat.ML cs.LG math.FA Submitted from: zhhaizh2(a)sysu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4439 or http://arXiv.org/abs/1101.4439

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Abstract of a paper by Guohui Song, Haizhang Zhang, Fred J. Hickernell
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Reproducing kernel Banach spaces with the l1 norm" by Guohui Song, Haizhang Zhang, Fred J. Hickernell. Abstract: Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given. Archive classification: stat.ML cs.LG math.FA Submitted from: zhhaizh2(a)sysu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4388 or http://arXiv.org/abs/1101.4388

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Abstract of a paper by Assaf Naor
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Sparse quadratic forms and their geometric applications (after Batson, Spielman and Srivastava)" by Assaf Naor. Abstract: We survey the work of Batson, Spielman and Srivastava on graph sparsification, and we describe some of its recently discovered geometric applications. Archive classification: math.FA Remarks: appeared as s\'eminaire Bourbaki expos\'e no. 1033, 2011 Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4324 or http://arXiv.org/abs/1101.4324

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Abstract of a paper by H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Multi-norms and the injectivity of $L^p(G)$" by H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden. Abstract: Let $G$ be a locally compact group, and take $p\in(1,\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces. Archive classification: math.FA Mathematics Subject Classification: 46H25, 43A20 Remarks: 27 pages Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4320 or http://arXiv.org/abs/1101.4320

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