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Unconditionality for $m$-homogeneous polynomials on $\ell _{\infty }^{n}$

Volume 232 / 2016

Andreas Defant, Pablo Sevilla-Peris Studia Mathematica 232 (2016), 45-55 MSC: Primary 46G25, 46B04; Secondary 46B28. DOI: 10.4064/sm8386-2-2016 Published online: 10 March 2016

Abstract

Let $\chi(m,n)$ be the unconditional basis constant of the monomial basis $z^\alpha $, $\alpha \in \mathbb{N}_0^n$ with $|\alpha|=m$, of the Banach space of all $m$-homogeneous polynomials in $n$ complex variables, endowed with the supremum norm on the $n$-dimensional unit polydisc $\mathbb{D}^n$. We prove that the quotient of $\sup_m\sqrt[m]{\sup_m\chi(m,n)}$ and $\sqrt{{n/\!\log n} }$ tends to $1$ as $n\to\infty$. This reflects a quite precise dependence of $\chi(m,n)$ on the degree $m$ of the polynomials and their number $n$ of variables. Moreover, we give an analogous formula for $m$-linear forms, a reformulation of our results in terms of tensor products, and as an application a solution for a problem on Bohr radii.

Authors

  • Andreas DefantInstitut für Mathematik
    Universität Oldenburg
    D-26111 Oldenburg, Germany
    e-mail
  • Pablo Sevilla-PerisInstituto Universitario de Matemática Pura y Aplicada
    Universitat Politècnica de València
    46022 Valencia, Spain
    e-mail

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