Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

Version 1 : Received: 28 December 2023 / Approved: 29 December 2023 / Online: 2 January 2024 (01:52:01 CET)

A peer-reviewed article of this Preprint also exists.

Yan, H.; Chen, J. Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces. Axioms 2024, 13, 108. Yan, H.; Chen, J. Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces. Axioms 2024, 13, 108.

Abstract

We investigate \texorpdfstring{$L_2$}{}-approximation problems in the worst case setting in the weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} with weights \texorpdfstring{$R_{d,\a,{\bm \ga}}$}{} under parameters \texorpdfstring{$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $1<\az_1\le \az_2\le \cdots$}{}. Several interesting weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. Multivariate approximation; information complexity; tractability; weighted Hilbert spacesWe discuss tractability of \texorpdfstring{$L_2$}{}-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on \texorpdfstring{$d$}{} and \texorpdfstring{$\va^{-1}$}{}. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and \texorpdfstring{$(t_1,t_2)$}{}-weak tractability (\texorpdfstring{$(t_1,t_2)$}{}-WT) for all \texorpdfstring{$t_1>1$}{} and \texorpdfstring{$t_2>0$}{} in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

Keywords

multivariate approximation; information complexity; tractability; weighted Hilbert spaces

Subject

Computer Science and Mathematics, Mathematics

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