Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Ohad Giladi and Assaf Naor
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "Improved bounds in the scaled Enflo type inequality for Banach spaces" by Ohad Giladi and Assaf Naor. Abstract: It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge 1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+ m\e/2)-f(x)}||_X^p] < C(p,X) m^p\sum_{j=1}^n\Avg_x[||f(x+e_j)-f(x)||_X^p], where the expectation is with respect to uniformly chosen x \in Z_m^n and \e \in \{-1,1\}^n, and C(p,X) is a constant that depends on p and the Rademacher type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained in [Mendel, Naor 2007]. The proof is based on an augmentation of the ``smoothing and approximation'' scheme, which was implicit in [Mendel, Naor 2007]. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 46B20, 51F99 Submitted from: giladi(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.4221 or http://arXiv.org/abs/1004.4221

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Abstract of a paper by M.E.Shirokov
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "Characterization of convex $\mu$-compact sets" by M.E.Shirokov. Abstract: The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class contains all compact sets as well as many noncompact sets widely used in applications. In this paper we give a characterization of a convex $\mu$-compact set in terms of properties of functions defined on this set. Namely, we prove that the class of convex $\mu$-compact sets can be characterized by continuity of the operation of convex closure of a function (= the double Fenchel transform) with respect to monotonic pointwise converging sequences of continuous bounded and of lower semicontinuous lower bounded functions. Archive classification: math.FA math.GM Citation: Russian Mathematical Surveys, 2008, 63:5 Remarks: 7 pages Submitted from: msh(a)mi.ras.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.3792 or http://arXiv.org/abs/1004.3792

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Abstract of a paper by Roman Vershynin
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "How close is the sample covariance matrix to the actual covariance matrix?" by Roman Vershynin. Abstract: Given a distribution in R^n, a classical estimator of its covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N = N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose the distribution is supported in a centered Euclidean ball of radius \sqrt{n}. We conjecture that the optimal sample size is N = O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson which states that N = O(n \log n) for distributions with finite second moment, and a recent result of Adamczak, Litvak, Pajor and Tomczak-Jaegermann which guarantees that N = O(n) for sub-exponential distributions. Archive classification: math.PR math.FA math.ST stat.TH Mathematics Subject Classification: 60H12, 60B20, 46B09 Remarks: 34 pages Submitted from: romanv(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.3484 or http://arXiv.org/abs/1004.3484

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Abstract of a paper by Daniel Pellegrino and Joedson Santos
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "A generalized unified Pietsch domination theorem and applications" by Daniel Pellegrino and Joedson Santos. Abstract: This paper has a twofold purpose. Firstly, we provide a new version of the Pietsch Domination Theorem that contains all the previous versions (to the best of our knowledge) as particular cases; our second goal is to characterize the arbitrary nonlinear mappings $f:X_{1}\times\cdots\times X_{n}\rightarrow Y$ that satisfy a quite natural Pietsch Domination-type theorem around a given point $(a_{1},...,a_{n})\in$ $X_{1}\times\cdots\times X_{n};$ as it will be shown, the new Pietsch Domination-type theorem plays a crucial role in this task. The characterization of such mappings lead to the idea of a kind of weighted summability for arbitrary mappings. Archive classification: math.FA Remarks: 12 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.2643 or http://arXiv.org/abs/1004.2643

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Abstract of a paper by Bernhard G. Bodmann, Peter G. Casazza, Vern I. Paulsen, and Darrin Speegle
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "Spanning and independence properties of frame partitions" by Bernhard G. Bodmann, Peter G. Casazza, Vern I. Paulsen, and Darrin Speegle. Abstract: We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. We also prove, assuming the Kadison-Singer conjecture is true, that this holds for infinite dimensional Hilbert spaces. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent, and the complement of each set to contain a number of disjoint, spanning sets. Archive classification: math.FA math.OA Submitted from: vern(a)math.uh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.2446 or http://arXiv.org/abs/1004.2446

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Abstract of a paper by Michael Dore and Olga Maleva
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "A compact universal differentiability set with Hausdorff dimension one" by Michael Dore and Olga Maleva. Abstract: We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space. Archive classification: math.FA math.CA Remarks: 11 pages Submitted from: o.maleva(a)bham.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.2151 or http://arXiv.org/abs/1004.2151

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Abstract of a paper by Hermann Pfitzner
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "The dual of a non-reflexive L-embedded Banach space contains $\ell^\infty$ isometrically." by Hermann Pfitzner. Abstract: See title. (A Banach space is said to be L-embedded if it is complemented in its bidual such that the norm between the two complementary subspaces is additive.) Archive classification: math.FA Remarks: accepted by Bull. Pol. Acad. Sci. Submitted from: Hermann.Pfitzner(a)univ-orleans.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.0203 or http://arXiv.org/abs/1004.0203

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Abstract of a paper by Philip A. H. Brooker
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Factorisation properties and space ideals associated with the Szlenk index" by Philip A. H. Brooker. Abstract: For $\alpha$ an ordinal, we study factorisation properties of the operator ideal $\mathscr{SZ}_\alpha$ of $\alpha$-Szlenk operators. We obtain quantitative factorisation results for Asplund operators in terms of the Szlenk index and a partial characterisation of those ordinals $\alpha$ for which $\mathscr{SZ}_\alpha$ has the factorisation property. Our investigations lead to the study of a class of space ideals defined in terms of a renorming property involving the Szlenk index. Archive classification: math.FA Submitted from: philip.brooker(a)anu.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5710 or http://arXiv.org/abs/1003.5710

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Abstract of a paper by Philip A. H. Brooker
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Direct sums and the Szlenk index" by Philip A. H. Brooker. Abstract: For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $\omega^\alpha$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $\varepsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums. Archive classification: math.FA Submitted from: philip.brooker(a)anu.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5708 or http://arXiv.org/abs/1003.5708

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Abstract of a paper by Philip A. H. Brooker
by alspach＠fourier.math.okstate.edu 15 Apr '10

15 Apr '10

This is an announcement for the paper "Operator ideals associated with the Szlenk index" by Philip A. H. Brooker. Abstract: For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$ consisting of all operators whose Szlenk index is an ordinal not exceeding $\omega^\alpha$. Our main result is that $\mathscr{SZ}_\alpha$ is a closed, injective, surjective operator ideal for each $\alpha$. We also study the relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known closed operator ideals. Archive classification: math.FA Submitted from: philip.brooker(a)anu.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1003.5706 or http://arXiv.org/abs/1003.5706

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