Banach December 2011

banach@mathdept.okstate.edu

2 participants
34 discussions

Abstract of a paper by Konstantin Storozhuk
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Strongly normal cones and the midpoint locally uniform rotundity" by Konstantin Storozhuk. Abstract: We give the method of construction of normal but not strongly normal positive cones in Banach space. Archive classification: math.FA Mathematics Subject Classification: 46B40 Remarks: 5 pages, 3 figures Submitted from: stork(a)math.nsc.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.1196 or http://arXiv.org/abs/1112.1196

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Abstract of a paper by D. Azagra
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Global approximation of convex functions" by D. Azagra. Abstract: We show that for every (not necessarily bounded) open convex subset $U$ of $\R^n$, every (not necessarily Lipschitz or strongly) convex function $f:U\to\R$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded sets to results on uniform approximation on unbounded sets, in such a way that not only convexity and $C^k$ smoothness, but also local Lipschitz constants, minimizers, order, and strict or strong convexity, are preserved. This transfer method is quite general and it can also be used to obtain new results on approximation of convex functions defined on Riemannian manifolds or Banach spaces. We also provide a characterization of the class of convex functions which can be uniformly approximated on $\R^n$ by strongly convex functions. Archive classification: math.FA math.CA math.DG Mathematics Subject Classification: 26B25, 41A30, 52A1, 46B20, 49N99, 58E99 Remarks: 16 pages Submitted from: dazagra(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.1042 or http://arXiv.org/abs/1112.1042

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Different forms of metric characterizations of classes of Banach spaces" by Mikhail I. Ostrovskii. Abstract: For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits uniformly bilipschitz embeddings of elements of the sequence H. Archive classification: math.FA math.CO math.MG Mathematics Subject Classification: Primary: 46B07, Secondary: 05C12, 46B85, 54E35 Remarks: Accepted for publication in Houston Journal of Mathematics Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.0801 or http://arXiv.org/abs/1112.0801

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Abstract of a paper by Yanqi Qiu
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "On the \text{UMD} constants for a class of iterated $L_p(L_q)$ spaces" by Yanqi Qiu. Abstract: Let $1 < p \neq q < \infty$ and $(D, \mu) = (\{\pm 1\}, \frac{1}{2} \delta_{-1} + \frac{1}{2}\delta_1)$. Define by recursion: $X_0 = \C$ and $X_{n+1} = L_p(\mu; L_q(\mu; X_n))$. In this paper, we show that there exist $c_1=c_1(p, q)>1$ and $ c_2 = c_2(p, q, s) > 1$, such that the $\text{UMD}_s$ constants of $X_n$'s satisfy $c_1^n \leq C_s(X_n) \leq c_2^n$ for all $1 < s < \infty$. Similar results will be showed for the analytic $\text{UMD}$ constants. We mention that the first super-reflexive non-$\text{UMD}$ Banach lattices were constructed by Bourgain. Our results yield another elementary construction of super-reflexive non-$\text{UMD}$ Banach lattices, i.e. the inductive limit of $X_n$, which can be viewed as iterating infinitely many times $L_p(L_q)$. Archive classification: math.FA Remarks: 18 pages Submitted from: yqi.qiu(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.0739 or http://arXiv.org/abs/1112.0739

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Abstract of a paper by Stephen Simons
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Linear $q$--positive sets and their polar subspaces" by Stephen Simons. Abstract: In this paper, we define a Banach SNL space to be a Banach space with a certain linear map from it into its dual, and we develop the theory of $q$--positive linear subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis–Browder theorem. Archive classification: math.FA Remarks: 11 pages Submitted from: simons(a)math.ucsb.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.0280 or http://arXiv.org/abs/1112.0280

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Abstract of a paper by Hossein Dehghan
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections" by Hossein Dehghan. Abstract: In this paper, a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space is proved by using metric projections. This theorem extends and improves the recent strong convergence theorem due to Matsushita and Takahashi [ Appl. Math. Comput. 196 (2008) 422-425] which was established for nonexpansive mappings. Archive classification: math.FA Mathematics Subject Classification: 47H09, 47H10 Submitted from: h_dehghan(a)iasbs.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.7107 or http://arXiv.org/abs/1111.7107

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Abstract of a paper by Taras Banakh and Ivan Hetman
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "A ``hidden'' characterization of approximatively polyhedral convex sets in Banach spaces" by Taras Banakh and Ivan Hetman. Abstract: For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\subset X$ and its metric component $H_C=\{A\in Conv_H(X):d_H(A,C)<\infty\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $\epsilon>0$ there is a polyhedral convex subset $P\subset X$ on Hausdorff distance $d_H(P,C)<\epsilon$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\subset X$; (3) the metric space $(H_C,d_H)$ is separable; (4) $H_C$ has density $dens(H_C)<\mathfrak c$; (5) $H_C$ does not contain a positively hiding convex set $P\subset X$. If the Banach space $X$ is finite-dimensional, then the conditions (1)--(5) are equivalent to: (6) $C$ is not positively hiding; (7) $C$ is not infinitely hiding. A convex subset $C\subset X$ is called {\em positively hiding} (resp. {\em infinitely hiding}) if there is an infinite set $A\subset X\setminus C$ such that $\inf_{a\in A}dist(a,C)>0$ (resp. $\sup_{a\in A}dist(a,C)=\infty$) and for any distinct points $a,b\in A$ the segment $[a,b]$ meets the set $C$. Archive classification: math.FA math.GN Mathematics Subject Classification: 46A55, 46N10, 52B05, 52A07, 52A27, 52A37 Remarks: 14 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.6708 or http://arXiv.org/abs/1111.6708

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Abstract of a paper by Valentin Ferenczi and Gilles Godefroy
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Tightness of Banach spaces and Baire category" by Valentin Ferenczi and Gilles Godefroy. Abstract: We prove several dichotomies on linear embeddings between Banach spaces. Given an arbitrary Banach space X with a basis, we show that the relations of isomorphism and bi-embedding are meager or co-meager on the Polish set of block-subspaces of X. We relate this result with tightness and minimality of Banach spaces. Examples and open questions are provided. Archive classification: math.FA Mathematics Subject Classification: 46B20, 54E52 Remarks: 13 pages Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.6444 or http://arXiv.org/abs/1111.6444

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Abstract of a paper by Ron Blei
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "The Grothendieck inequality revisited" by Ron Blei. Abstract: The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is a representation of the inner product in a Hilbert space by an integral with uniformly bounded and continuous integrands. The Parseval-like formula is obtained by iterating the usual Parseval formula in a framework of harmonic analysis on dyadic groups. A modified construction implies a similar integral representation of the dual action between $l^p$ and $l^q$, \ $\frac{1}{p} + \frac{1}{q} = 1$. Variants of the Grothendieck inequality are derived in higher dimensions. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n.$ Multilinear Parseval-like formulas are obtained, extending the bilinear formula. The resulting formulas yield multilinear extensions of the bilinear Grothendieck inequality, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space, within a class of functionals whose kernels are supported by fractional Cartesian products. Archive classification: math.FA Submitted from: blei(a)math.uconn.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.7304 or http://arXiv.org/abs/1111.7304

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Abstract of a paper by Andrzej Wisnicki
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "On the fixed points of nonexpansive mappings in direct sums of Banach spaces" by Andrzej Wisnicki. Abstract: We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then a direct sum of X and Y, with respect to a strictly monotone norm, has the weak fixed point property. The result is new even if Y is a finite-dimensional space. Archive classification: math.FA Remarks: 9 pages. To appear, Studia Mathematica Submitted from: awisnic(a)golem.umcs.lublin.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.6965 or http://arXiv.org/abs/1111.6965

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Banach December 2011