Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Ondrej Kurka
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Genericity of Fr\'echet smooth spaces" by Ondrej Kurka. Abstract: If a separable Banach space contains an isometric copy of every separable reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fr\'echet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04, 46B20, Secondary 46B15, 54H05 Remarks: 34 pages Submitted from: kurka.ondrej(a)seznam.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5726 or http://arXiv.org/abs/1109.5726

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Abstract of a paper by Anna Kaminska and Damian Kubiak
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "On the dual of Ces\`aro function space" by Anna Kaminska and Damian Kubiak. Abstract: The goal of this paper is to present an isometric representation of the dual space to Ces\`aro function space $C_{p,w}$, $1<p<\infty$, induced by arbitrary positive weight function $w$ on interval $(0,l)$ where $0<l\leqslant\infty$. For this purpose given a strictly decreasing nonnegative function $\Psi$ on $(0,l)$, the notion of essential $\Psi$-concave majorant $\hat f$ of a measurable function $f$ is introduced and investigated. As applications it is shown that every slice of the unit ball of the Ces\`aro function space has diameter 2. Consequently Ces\`aro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B42, 46B22 Remarks: 15 pages Submitted from: dmkubiak(a)memphis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5400 or http://arXiv.org/abs/1109.5400

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Abstract of a paper by Qingying Bu, Gerard Buskes, Alexey I. Popov, Adi Tcaciuc, and Vladimir G. Troitsky
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square" by Qingying Bu, Gerard Buskes, Alexey I. Popov, Adi Tcaciuc, and Vladimir G. Troitsky. Abstract: We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice $E$ with itself and the 2-concavification of~$E$. Archive classification: math.FA Mathematics Subject Classification: 46B42, 46M05, 46B40, 46B45 Remarks: 18 pages Submitted from: troitsky(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5380 or http://arXiv.org/abs/1109.5380

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Abstract of a paper by Casey Kelleher, Daniel Miller, Trenton Osborn, and Anthony Weston
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Strongly non embeddable metric spaces" by Casey Kelleher, Daniel Miller, Trenton Osborn, and Anthony Weston. Abstract: Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(\mathfrak{Z}, \zeta)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $\mathbb{Z}_\omega=\bigoplus_{\aleph_0}\mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p \geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z \to \ell_{p}$. Archive classification: math.FA math.GN Mathematics Subject Classification: 46C05, 46T99 Remarks: 10 pages Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5300 or http://arXiv.org/abs/1109.5300

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Abstract of a paper by Sergey Bobkov and Mokshay Madiman
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures" by Sergey Bobkov and Mokshay Madiman. Abstract: We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of $M$-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Pl\"unnecke-Ruzsa inequalities from additive combinatorics. Archive classification: math.FA math.PR Remarks: 28 pages, revised version of a document submitted in October 2010 Submitted from: mokshay.madiman(a)yale.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5287 or http://arXiv.org/abs/1109.5287

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Abstract of a paper by Konrad J. Swanepoel and Rafael Villa
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "Maximal equilateral sets" by Konrad J. Swanepoel and Rafael Villa. Abstract: A subset of a normed space X is called equilateral if the distance between any two points is the same. Let m(X) be the smallest possible size of an equilateral subset of X maximal with respect to inclusion. We first observe that Petty's construction of a d-dimensional X of any finite dimension d >= 4 with m(X)=4 can be generalised to show that m(X\oplus_1\R)=4 for any X of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that both m(\ell_p) and m(\ell_p^d) are finite and bounded above by a function of p, for all 1 <= p < 2. Also, for all p in [1,\infty) and natural numbers d there exists c=c(p,d) > 1 such that m(X) <= d+1 for all d-dimensional X with Banach-Mazur distance less than c from \ell_p^d. Using Brouwer's fixed-point theorem we show that m(X) <= d+1 for all d-\dimensional X with Banach-Mazur distance less than 3/2 from \ell_\infty^d. A graph-theoretical argument furthermore shows that m(\ell_\infty^d)=d+1. The above results lead us to conjecture that m(X) <= 1+\dim X. Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: 46B20 (Primary), 46B85, 52A21, 52C17 (Secondary) Remarks: 15 pages Submitted from: konrad.swanepoel(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5063 or http://arXiv.org/abs/1109.5063

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Abstract of a paper by Adriano Thiago Bernardino, Daniel Pellegrino and Juan B. Seoane-Sepulveda
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "Multiple $(p;q;r)$-summing polynomials and multilinear operators" by Adriano Thiago Bernardino, Daniel Pellegrino and Juan B. Seoane-Sepulveda. Abstract: The concept of absolutely $(p;q;r)$-summing linear operators is due to A. Pietsch; it is a natural extension of the classical notion of absolutely $(p;q)$-summing operators. Very recently D. Achour introduced the concept of absolutely $(p;q;r)$-summing multilinear mappings. In this paper we obtain some properties of this class and show that the polynomial version of this notion is neither coherent nor compatible (according to the definition of Carando, Dimant, and Muro). Here we shall provide an alternative approach that generates coherent and compatible ideals. 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/1109.4898 or http://arXiv.org/abs/1109.4898

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Abstract of a paper by Parcet Javier, Soria Fernando, and Xu Quanhua
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "On the growth of vector-valued Fourier series" by Parcet Javier, Soria Fernando, and Xu Quanhua. Abstract: We prove the 'little Carleson theorem' on the growth of Fourier series for functions taking values in a UMD Banach space. Archive classification: math.CA math.FA Submitted from: javier.parcet(a)icmat.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.4313 or http://arXiv.org/abs/1109.4313

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Abstract of a paper by Sergo A. Episkoposian
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "On greedy algorithms with respect to generalized Walsh system" by Sergo A. Episkoposian. Abstract: In this paper we proof that there exists a function f(x) belongs to L^1[0,1] such that a greedy algorithm with regard to generalized Walsh system does not converge to f(x) in L^1[0,1] norm, i.e. the generalized Walsh system is not a quasi-greedy basis in its linear span L^1[0,1]. Archive classification: math.FA Mathematics Subject Classification: 42C10, 46E30 Submitted from: sergoep(a)ysu.am The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.3806 or http://arXiv.org/abs/1109.3806

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Abstract of a paper by Sergo A. Episkoposian
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "On the existence of universal series by trigonometric system" by Sergo A. Episkoposian. Abstract: In this paper we prove the following: let $\omega(t)$ be a continuous function, increasing in $[0,\infty)$ and $\omega(+0)=0$. Then there exists a series of the form $$\sum_{k=-\infty}^\infty C_ke^{ikx} \ \ with \ \ \sum_{k=-\infty}^\infty C^2_k \omega(|C_k|)<\infty ,\ \ C_{-k}=\overline{C}_k, \eqno$$ with the following property: for each $\varepsilon>0$ a weighted function $\mu(x), 0<\mu(x) \le1, \left | \{ x\in[0,2\pi]: \mu(x)\not =1 \} \right | <\varepsilon $ can be constructed, so that the series is universal in the weighted space $L_\mu^1[0,2\pi]$ with respect to rearrangements. Archive classification: math.FA Mathematics Subject Classification: 42A20 Submitted from: sergoep(a)ysu.am The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.3805 or http://arXiv.org/abs/1109.3805

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