Kolmogorov \(n\)-width of some finite-dimensional sets in a mixed norm
type: Item
instance of: scholarly article
title: Kolmogorov \(n\)-width of some finite-dimensional sets in a mixed norm
zbMATH Open document ID: 0865.41022
DOI: 10.1007/BF02306188
author: Eh. M. Galeev
published in: Mathematical Notes
publication date: 1997-07-08T00:00:00Z
review text: Denote by \(V^n_k\) the convex hull in \(\mathbb{R}^n\) of the points having \(k\) coordinates equal to \(\pm 1\) and the remaining equal to zero, and denote \(V_{k,\infty}^{n,m} =V^k_n\times \cdots \times V_n^k\). For \(x\in\mathbb{R}^{nm}\), let \[ |x|_{l^{n,m}_{p,q}} = \left(\sum^m_{s=1} \Bigl(\sum_{k\in\Delta_s} |x_k|^p \Bigr)^{q/p} \right)^{1/q} \quad \text{with} \quad \Delta_s= \bigl\{k\in N;(s-1) n<k<sn\bigr\}. \] The following asymptotics for the Kolmogorov width \(d_N(V^{n,m}_{k,\infty}, l^{n,m}_{p,q})=d\) are determined: \(d\asymp k^{1/p} m^{1/q}\) provided \(1<q \leq\infty\), \(N<nm/2\), \(1\leq k\leq n\), \(p=2\) or \(1<p\leq \min(q,2)\).
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A46
zbMATH DE Number: 898918
zbMATH Keywords: Kolmogorov width
reviewed by: Alois Kufner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 898918