Optimal recovery of operators in function \(L\)-spaces
type: Item
instance of: scholarly article
title: Optimal recovery of operators in function \(L\)-spaces
zbMATH Open document ID: 1488.46070
author: Yanyan Li
published in: Analysis Mathematica
publication date: 2021-06-29T00:00:00Z
review text: A semi-linear space is a linear space lacking the existence of the opposite elements and of the distributive law \((\lambda+\mu)x= \lambda x+\mu x\). An \(L\)-space is a semilinear space \(X\) endowed with a metric \(h_X\) such that (i)\; \(h_X(\alpha x,\alpha y)=\vert \alpha \vert h_X(x,y)\) and (ii)\; \(h_X(x+z,y+z)\le h_X(x,y)\) for all \(x,y,z\in X\) and \(\alpha \in\mathbb{R}\). The \(L\)-space \((X,h_X)\) is called isotropic if (ii) holds with equality. An element \(x\in X\) is called convex if \((\alpha+\beta)x= \alpha x+\beta x\) for all \(\alpha,\beta\ge 0\). A typical example is the space \(\Omega(X)\) of all nonempty closed bounded subsets of a normed linear space \(X\) with the Hausdorff metric. In this case, the convex elements are the convex sets in \(\Omega(X)\). \par The paper is concerned with the problem of optimal recovery of the operators on the classes \(H^\omega(T,X)\) of continuous functions defined on a metric compact \(T\), taking values in an \(L\)-space \(X\) and having a given majorant \(\omega(t) \) of their modulus of continuity. The operators are recovered based on the values (known with errors) of the functions on a finite set of points.\par From the text: ``The consideration of operators defined on function \(L\)-spaces allows us to include many important operators acting in the spaces of functions with values in Banach spaces (in particular in the spaces of random processes), spaces of set-valued functions and functions with fuzzy values, and obtain new results on the optimal recovery of such operators.''
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 46A19
zbMATH DE Number: 7364735
zbMATH Keywords: optimal recovery
zbMATH Keywords: \(L\)-space
zbMATH Keywords: modulus of continuity
zbMATH Keywords: fuzzy set
zbMATH Keywords: set-valued function
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: s10476-021-0065-y
OpenAlex ID: W3123131469
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DOI: 10.1007/S10476-021-0065-Y
description: scientific article; zbMATH DE number 7364735