Metric entropy of integration operators and small ball probabilites for the Brownian sheet
type: Item
instance of: scholarly article
title: Metric entropy of integration operators and small ball probabilites for the Brownian sheet
zbMATH Open document ID: 0945.47022
author: Werner Linde
author: Thomas Kühn
author: Thomas Dunker
author: M. A. Lifshits
published in: Journal of Approximation Theory
publication date: 2000-10-04T00:00:00Z
review text: The authors consider the \(d-\)dimensional integration operator \( T_{d}:L_{2}([0,1]^{d})\rightarrow C([0,1]^{d})\) defined as \[ T_{d}f(x_{1},x_{2,}...,x_{d}):=\int_{0}^{1}.... \int_{0}^{1}f(y_{1},...,y_{d})dy_{1}...dy_{d}, f\in L_{2}([0,1]^{d}). \] The main result concerning \(T_{d}\) is the following estimate for its Kolmogorov and entropy numbers. {Theorem 1.} There exists constants \(C_{1}\) and \(C_{2\text{ }}\) such that \[ d_{k}(T_{d})\leq \frac{C_{1}}{k}(1+\log k)^{d-1/2}\text{ and } e_{k}(T_{d})\leq \frac{C_{2}}{k}(1+\log k)^{d-1/2} \] hold for all \(k\geq 1.\)
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 1389110
zbMATH Keywords: metric entropy
zbMATH Keywords: small ball probabilities
zbMATH Keywords: Brownian sheet
zbMATH Keywords: \(d\)-dimensional integration operator
zbMATH Keywords: Kolmogorov and entropy numbers
reviewed by: Vladimir Rabinovich
MaRDI profile type: Publication
full work available at URL: jath.1999.3354
OpenAlex ID: W2053302979
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DOI: 10.1006/JATH.1999.3354
description: scientific article; zbMATH DE number 1389110