Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by David Alonso-Gutierrez and Joscha Prochno Title: Estimating support functions of random polytopes via Orlicz norms
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "Estimating support functions of random polytopes via Orlicz norms" by David Alonso-Gutierrez and Joscha Prochno. Abstract: We study the expected value of support functions of random polytopes in a certain direction, where the random polytope is given by independent random vectors uniformly distributed in an isotropic convex body. All results are obtained by an utterly novel approach, using probabilistic estimates in connection with Orlicz norms that were not used in this connection before. Archive classification: math.FA Mathematics Subject Classification: Primary 52A22, Secondary 52A23, 05D40, 46B09 Submitted from: prochno(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1205.2023 or http://arXiv.org/abs/1205.2023

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Abstract of a paper by Tuomas P. Hytonen and Michael T. Lacey
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "Pointwise convergence of vector-valued Fourier series" by Tuomas P. Hytonen and Michael T. Lacey. Abstract: We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions. Archive classification: math.FA math.CA Mathematics Subject Classification: 42B20, 42B25 Remarks: 26 pages Submitted from: tuomas.hytonen(a)helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1205.0261 or http://arXiv.org/abs/1205.0261

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Abstract of a paper by Szymon Glab, Pedro L. Kaufmann and Leonardo Pellegrini
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "Spaceability and algebrability of sets of nowhere integrable functions" by Szymon Glab, Pedro L. Kaufmann and Leonardo Pellegrini. Abstract: We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269], and that it is strongly $\mathfrak{c}$-algebrable. We prove strong $\mathfrak{c}$-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesgue-nowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not $1$-algebrable. We also show that there exists an infinite dimensional vector space $S$ of differentiable functions such that each element of the $C([0,1])$-closure of $S$ is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from [V. I. Gurariy, \textit{Subspaces and bases in spaces of continuous functions (Russian),} Dokl. Akad. Nauk SSSR, \textbf{167} (1966), 971--973]. Archive classification: math.FA Remarks: accepted on 2011 Submitted from: leoime(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.6404 or http://arXiv.org/abs/1204.6404

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Abstract of a paper by Joscha Prochno
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "The embedding of 2-concave Musielak-Orlicz spaces into L_1 via l_2-matrix-averages" by Joscha Prochno. Abstract: In this note we prove that $\frac{1}{n!} \sum_{\pi} ( \sum_{i=1}^n |x_i a_{i,\pi(i)} |^2)^{\frac{1}{2}}$ is equivalent to a Musielak-Orlicz norm $\norm{x}_{\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of strictly 2-concave Musielak-Orlicz spaces into L_1. Archive classification: math.FA Mathematics Subject Classification: 46B03, 05A20, 46B45 Submitted from: prochno(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.6030 or http://arXiv.org/abs/1204.6030

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Abstract of a paper by Joscha Prochno and Carsten Schuett
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "Combinatorial inequalities and subspaces of L1" by Joscha Prochno and Carsten Schuett. Abstract: Let M and N be Orlicz functions. We establish some combinatorial inequalities and show that the product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1 if M and N are "separated" by a function t^r, 1<r<2. Archive classification: math.FA math.CO Mathematics Subject Classification: 46B03, 05A20, 46B45, 46B09 Submitted from: prochno(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.6025 or http://arXiv.org/abs/1204.6025

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Abstract of a paper by G. Botelho, D. Pellegrino, P. Rueda, J. Santos and J.B. Seoane-Sepulveda
by alspach＠math.okstate.edu 11 Jun '12

11 Jun '12

This is an announcement for the paper "When is the Haar measure a Pietsch measure for nonlinear mappings?" by G. Botelho, D. Pellegrino, P. Rueda, J. Santos and J.B. Seoane-Sepulveda. Abstract: We show that, as in the linear case, the normalized Haar measure on a compact topological group $G$ is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of $C(G)$. This answers a question posed to the authors by J. Diestel. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section some problems are proposed. Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.5621 or http://arXiv.org/abs/1204.5621

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Abstract of a paper by Tao Mei
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "A universal $H_1$-BMO duality theory for semigroups of operators" by Tao Mei. Abstract: Let (M,\mu) be a sigma-finite measure space. Let (T_t) be a semigroup of positive preserving maps on (M,\mu) with standard assumptions. We prove a $H_1$-BMO duality theory with assumptions only on the semigroup of operators. The H1's are defined by square functions of P. A. Meyer's gradient form. The formulation of the assumptions does not rely on any geometric/metric property of M nor on the kernel of the semigroups of operators. Our main results extend to the noncommutative setting as well, e.g. the case where $L_\infty(M,\mu)$ is replaced by von Neuman algebras with a semifinite trace. We also prove a Carlson embedding theorem for semigroups of operators. Archive classification: math.CA math.FA math.OA Mathematics Subject Classification: 46L51 42B25 46L10 47D06 Remarks: 22 pages Submitted from: mei(a)wayne.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.4424 or http://arXiv.org/abs/1005.4424

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Abstract of a paper by Daniel Pellegrino
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Sharp coincidences for absolutely summing multilinear operators" by Daniel Pellegrino. Abstract: In this note we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method. Archive classification: math.FA Remarks: This note is part of the author's thesis which is being written for The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.5411 or http://arXiv.org/abs/1204.5411

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Abstract of a paper by Wayne Lawton
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Spectral envelopes - A preliminary report" by Wayne Lawton. Abstract: The spectral envelope S(F) of a subset of integers is the set of probability measures on the circle group that are weak star limits of squared moduli of trigonometric polynomials with frequencies in F. Fourier transforms of these measures are positive and supported in F - F but the converse generally fails. The characteristic function chiF of F is a binary sequence whose orbit closure gives a symbolic dynamical system O(F). Analytic properties of S(F) are related to dynamical properties of chiF. The Riemann-Lebesque lemma implies that if chiF is minimal, then S(F) is convex and hence S(F) is the closure of the convex hull of its extreme points Se(F). In this paper we (i) review the relationship between these concepts and the special case of the still open 1959 Kadison-Singer problem called Feichtinger's conjecture for exponential functions, (ii) partially characterize of elements in Se(F), for minimal chiF, in terms of ergodic properties of (O(F),lambda) where lambda is a shift invariant probability measure whose existence in ensured by the 1937 Krylov-Bogoyubov theorem, (iii) refine previous numerical studies of the Morse-Thue minimal binary sequence by exploiting a new MATLAB algorithm for computing smallest eigenvalues of 4,000,000 x 4,000,000 matrices, (iv) describe recent results characterizing S(F) for certain Bohr sets F related to quasicrystals, (v) extend these concepts to general discrete groups including those with Kazhdan's T-property, such as SL(n,Z), n > 2, which can be characterized by several equivalent properties such as: any sequence of positive definite functions converging to 1 uniformly on compact subsets converges uniformly. This exotic property may be useful to construct a counterexample to the generalization of Feichtinger's conjecture and hence to provide a no answer to the question of Kadison and Singer whcih they themselves tended to suspect. Archive classification: math.FA Mathematics Subject Classification: 37B10, 42A55, 43A35 Remarks: To appear in Proceedings the Annual Meeting in Mathematics, Bangkok, The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.4904 or http://arXiv.org/abs/1204.4904

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Abstract of a paper by Laurent W. Marcoux, Alexey I. Popov, and Heydar Radjavi
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "On almost-invariant subspaces and approximate commutation" by Laurent W. Marcoux, Alexey I. Popov, and Heydar Radjavi. Abstract: A closed subspace of a Banach space $\cX$ is almost-invariant for a collection $\cS$ of bounded linear operators on $\cX$ if for each $T \in \cS$ there exists a finite-dimensional subspace $\cF_T$ of $\cX$ such that $T \cY \subseteq \cY + \cF_T$. In this paper, we study the existence of almost-invariant subspaces of infinite dimension and codimension for various classes of Banach and Hilbert space operators. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if $T$ is an operator on a separable Hilbert space and if $TP-PT$ has finite rank for all projections $P$ in a given maximal abelian self-adjoint algebra $\fM$ then $T=M+F$ where $M\in\fM$ and $F$ is of finite rank. Archive classification: math.FA math.OA Mathematics Subject Classification: 47A15, 47A46, 47B07, 47L10 Submitted from: a4popov(a)uwaterloo.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.4621 or http://arXiv.org/abs/1204.4621

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