Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1≥γ1≥γ2≥⋯≥0 and 1<α1≤α2≤⋯. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms...

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Bibliographic Details
Published inAxioms Vol. 13; no. 2; p. 108
Main Authors Yan, Huichao, Chen, Jia
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.02.2024
Subjects
Algorithms
Approximation
Approximation theory
Continuity (mathematics)
Criteria
Eigenvalues
Errors
Hilbert space
information complexity
Mathematical analysis
Mathematical research
Methods
multivariate approximation
Polynomials
tractability
weighted Hilbert spaces
Weighting (Statistics)
China
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Summary:We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1≥γ1≥γ2≥⋯≥0 and 1<α1≤α2≤⋯. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε−1. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1>1 and t2>0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13020108