Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes
type: Item
instance of: scholarly article
title: Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes
zbMATH Open document ID: 1066.47017
author: Werner Linde
published in: Journal of Approximation Theory
publication date: 2004-08-10T00:00:00Z
review text: The author studies the compactnesss properties of the Riemann-Liouville fractional integral operator \(R_\alpha\) from \(L_{2}[0,1]\) into \(C(K)\), where \(K\) is any of certain compact subsets of [0,1]. The degree of compactness of \(R_\alpha\) is measured using the Kolmogorov numbers \(d_ n(K)\) and the entropy numbers \(\varepsilon_ n(K)\) of the underlying set \(K\). A relationship between these two numbers is established and for the sake of completeness, we reproduce one of the results obtained: Suppose that \(\alpha > \frac12\) and let \( K \subset [0,1]\) be a compact set such that for some \(\lambda \geq 1\) \[ \varepsilon_{m}(K) \leq \lambda\cdot \varepsilon_{2m}(K), \quad m\in {\mathbb N}. \] Then it follows that \[ d_ n (R_\alpha : L_ 2 [0,1] \to C(K)) \geq c\cdot m^{-\frac1{2}}\cdot \epsilon_ m (K)^{\alpha-\frac12}, \] where \(c > 0\) only depends on \(\alpha\) and \(\lambda\). Applications of the obtained results to the small ball behaviour of certain Gaussian stochastic processes such as fractional Brownian motion or Riemann-Liouville processes indexed by small (fractal) sets are given.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2085994
zbMATH Keywords: fractional integration
zbMATH Keywords: Kolmogorov numbers
zbMATH Keywords: entropy numbers
zbMATH Keywords: fractal sets
zbMATH Keywords: small deviation
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: degree of compactness
MaRDI profile type: Publication
full work available at URL: j.jat.2004.04.011
OpenAlex ID: W1975074544
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DOI: 10.1016/J.JAT.2004.04.011
description: scientific article; zbMATH DE number 2085994