April 2008 - Banach - mathdept.okstate.edu

Abstract of a paper by Christian Le Merdy and Fedor Sukochev
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Rademacher averages on noncommutative symmetric spaces" by Christian Le Merdy and Fedor Sukochev. Abstract: Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let $(\varepsilon_k)_k$ be a Rademacher sequence, on some probability space $\Omega$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages $\sum_k \varepsilon_k\otimes x_k$ as elements of the noncommutative function space $E(L^\infty(\Omega)\otimes M)$ and study estimates for their norms $\Vert \sum_k \varepsilon_k \otimes x_k\Vert_E$ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of $\Vert (\sum y_k^*y_k)^{\frac{1}{2}}\Vert + \Vert (\sum z_k z_k^*)^{\frac{1}{2}}\Vert$ over all $y_k,z_k$ in E(M) such that $x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M). Archive classification: math.FA math.OA Mathematics Subject Classification: 46L52; 46M35; 47L05 The source file(s), KHTot.tex: 72248 bytes, is(are) stored in gzipped form as 0803.4404.gz with size 20kb. The corresponding postcript file has gzipped size 152kb. Submitted from: clemerdy(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4404 or http://arXiv.org/abs/0803.4404 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4404 or in gzipped form by using subject line get 0803.4404 to: math(a)arXiv.org.

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Abstract of a paper by Th. Schlumprecht and N. Sivakumar
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian" by Th. Schlumprecht and N. Sivakumar. Abstract: Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a Riesz basis ({\it i.e.,\/} unconditionalbasis) for $L_2[-\pi,\pi]$. Given a function $f\in L_2(\mathbb R)$ whose Fourier transform is zero almost everywhere outside the interval $[-\pi,\pi]$, there is a unique square-summable sequence $(a_j:j\in\mathbb Z)$, depending on $\lambda$ and $f$, such that the function$$I_\lambda(f)(x):=\sum_{j\in\mathbb Z}a_je^{-\lambda(x-x_j)^2}, \qquad x\in\mathbb R, $$ is continuous and square integrable on $(-\infty,\infty)$, and satisfies the interpolatory conditions $I_\lambda (f)(x_j)=f(x_j)$, $j\in\mathbb Z$. It is shown that $I_\lambda(f)$ converges to $f$ in $L_2(\mathbb R)$, and also uniformly on $\mathbb R$, as $\lambda\to0^+$. A multidimensional version of this result is also obtained. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on $\ell_p(\mathbb Z)$ for every $p\in[1,\infty]$. Archive classification: math.CA math.FA Mathematics Subject Classification: 41A05 46E15 The source file(s), scsi1_5.tex: 93892 bytes, is(are) stored in gzipped form as 0803.4344.gz with size 27kb. The corresponding postcript file has gzipped size 165kb. Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4344 or http://arXiv.org/abs/0803.4344 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4344 or in gzipped form by using subject line get 0803.4344 to: math(a)arXiv.org.

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Abstract of a paper by Anton R. Schep
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Products and factors of Banach function spaces" by Anton R. Schep. Abstract: Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F. Archive classification: math.FA Mathematics Subject Classification: 46E30; 47B38 Remarks: 16 pages The source file(s), product-bfs.bbl: 4503 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4336 or http://arXiv.org/abs/0803.4336 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4336 or in gzipped form by using subject line get 0803.4336 to: math(a)arXiv.org.

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Abstract of a paper by Yauhen Radyna, Yakov Radyno, and Anna Sidorik
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers" by Yauhen Radyna, Yakov Radyno, and Anna Sidorik. Abstract: We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator. Archive classification: math.FA Mathematics Subject Classification: 46C15, 43A25 Citation: Yauhen Radyna, Yakov Radyno, Anna Sidorik, Characterizing Hilbert The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.3646 or http://arXiv.org/abs/0803.3646 or by email in unzipped form by transmitting an empty message with subject line uget 0803.3646 or in gzipped form by using subject line get 0803.3646 to: math(a)arXiv.org.

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