Norm estimates for random polynomials on the scale of Orlicz spaces
type: Item
instance of: scholarly article
title: Norm estimates for random polynomials on the scale of Orlicz spaces
zbMATH Open document ID: 1400.46040
DOI: 10.1215/17358787-0000006X
author: Andreas Defant
author: Mieczysław Mastyło
published in: Banach Journal of Mathematical Analysis
publication date: 2017-03-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1485572421
review text: The authors prove an upper bound for the uniform norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane-Salem-Zygmund inequality, see Theorem 4, Chapter 6 in [\textit{J.-P. Kahane}, Some random series of functions. 2nd ed. Cambridge: Cambridge University Press (1985; Zbl 0571.60002)] and its extension in [\textit{F. Bayart}, Q. J. Math. 63, No.\,1, 21--39 (2012; Zbl 1248.46033)]. The main idea is the following definition: Let \(\psi : \mathbb N_{>1} \times \mathbb N \rightarrow [1,\infty )\) and \(\phi : \mathbb N \rightarrow [1,\infty )\) be increasing functions. Then, a Banach sequence space \(E\) (modelled on \(\mathbb N\)) is said to be \((\psi,\phi)\)-admissible, provided that there is a constant \(C>0\) such that, for every choice of scalars \((c_\mathbf{j})_{\mathbf{j} \in \mathcal{J}(m,n)}\) with \(m>1\) and \(n \geq 1\), and functionals \(x_1^\ast, \dots, x_n^\ast \in X^\ast\) on an \(n\)-dimensional Banach space \(X\), there exists a choice of signs \(\varepsilon _\mathbf{j} = \pm 1\), \({\mathbf{j}\in \mathcal{J}(m,n)}\), such that \[ \sup_{z \in B_X} \left|\sum_{\mathbf{j}\in \mathcal{J}(m,n)} \varepsilon _\mathbf{j} c_\mathbf{j} x_\mathbf{j}^\ast(z) \right| \leq C \psi(m,n) \sup_{\mathbf{j} \in \mathcal{J}(m,n)} \frac{|c_{\mathbf{j}}|}{\phi(|[\mathbf{j}]|)} \sup_{z \in B_X} \left\|\sum_{k=1}^n x_k^\ast(z)e_k\right\|_E^m. \] Here, \[ \begin{aligned} \mathcal{M}(m,n) &= \{\mathbf{i} = (i_1,\dots,i_m): 1\leq i_k \leq n\} = \{1,\dots,n\}^m,\\ \mathcal{J}(m,n) &= \{\mathbf{j} = (j_1,\dots,j_m) \in \mathcal{M}(m,n): 1\leq j_1 \leq j_2 \leq \cdots \leq j_m \leq n\}.\end{aligned} \] Moreover, indices \(\mathbf{i},\mathbf{j} \in \mathcal{M}(m,n)\) are called equivalent if there is a permutation \(\sigma \) such that \((i_1,\dots,i_m)=(j_{\sigma(1)},\dots,j_{\sigma(m)})\). The equivalence class of \(\mathbf{i}\) is denoted by \([\mathbf{i}]\) and \(|[\mathbf{i}]|\) denot…
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 47H60
Mathematics Subject Classification ID: 46B45
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 6694357
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: interpolation spaces
zbMATH Keywords: Orlicz spaces
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6694357