15 research outputs found

    Representable Functions on the Unit Ball of a Banach Space

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    In this paper we treat the problem of integral representation of analytic functions over the unit ball of a complex Banach space X using the theory of abstract Wiener spaces. We define the class of representable functions on the unit ball of X and prove that this set of functions is related with the classes of integral k-homogeneous polynomials, integral holomorphic functions and also with the set of Lp-representable functions on a Banach space.Fil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentin

    Non-linear Plank Problems and polynomial inequalities

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    We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works when the number of polynomials is large.Comment: 19 page

    Dynamics of non-convolution operators and holomorphy types

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    In this article we study the hypercyclic behavior of non-convolution operators defined on spaces of analytic functions of different holomorphy types over Banach spaces. The operators in the family we analyze are a composition of differentiation and composition operators, and are extensions of operators in H(C) studied by Aron and Markose in 2004. The dynamics of this class of operators, in the context of one and several complex variables, was further investigated by many authors. It turns out that the situation is somewhat different and that some purely infinite dimensional difficulties appear. For example, in contrast to the several complex variable case, it may happen that the symbol of the composition operator has no fixed points and still, the operator is not hypercyclic. We also prove a Runge type theorem for holomorphy types on Banach spaces.Fil: Muro, Luis Santiago Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaFil: Savransky, Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

    On the measure of polynomials attaining maxima on a vertex

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    We calculate the probability that a k-homogeneous polynomial in n variables attain a local maximum on a vertex in terms of the “sharpness” of the vertex, and then study the dependence of this measure on the growth of dimension and degree. We find that the behavior of vertices with orthogonal edges is markedly different to that of sharper vertices. If the degree k grows with the dimension n, the probability that a polynomial attain a local maximum tends to 1/2, but for orthogonal edges the growth-rate of k must be larger than nlnn, while for sharper vertices a growth-rate larger than lnn will suffice.Fil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin

    A probabilistic Approach to polynomial Inequalities

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    We define a probability measure on the space of polynomials over Rn in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities. Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of n 1 tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.Fil: Pinasco, Damian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Asymptotic estimates for the largest volume ratio of a convex body

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    The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where the sup runs over all the convex bodies L. We prove the following sharp lower bound: c √n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n/log log(n). We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Merzbacher, Diego Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentin

    The Minimal Volume of Simplices Containing a Convex Body

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    Let K⊂ Rn be a convex body with barycenter at the origin. We show there is a simplex S⊂ K having also barycenter at the origin such that (vol(S)vol(K))1/n≥cn, where c> 0 is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if K is in isotropic position, we present a method to find centered simplices verifying the above bound that works with extremely high probability. By duality, given a convex body K⊂ Rn we show there is a simplex S enclosing Kwith the same barycenter such that(vol(S)vol(K))1/n≤dn,for some absolute constant d> 0. Up to the constant, the estimate cannot be lessened.Fil: Galicer, Daniel Eric. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Merzbacher, Diego Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Linear and bilinear operators and their zero-sets

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    We study the extent to which several classical results relating linear or multilinear forms and their zero-sets can be generalised to linear or bilinear operators with values in Rn. We find some analogues of the classical theorems, and also some restrictions.Fil: Pinasco, Damian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Tchakaloff’s theorem and k-integral polynomials in Banach spaces

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    Tchakaloff’s theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff’s theorem in an infinite-dimensional Banach space E: a weak form valid when E has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space E.Fil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin

    Orthant probabilities and the attainment of maxima on a vertex of a simplex

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    We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree k>4, the probability that a k-homogeneous polynomial in n variables attains a local constrained maximum on a vertex of the n-dimensional simplex tends to one as the dimension n grows. The bounds we obtain for the orthant probabilities are tight up to log⁡(n) factors.Fil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; ArgentinaFil: Smucler, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; ArgentinaFil: Zalduendo, Ignacio Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin
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