Functiones et Approximatio Commentarii Mathematici

Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

Peter Kritzer, Helene Laimer, and Friedrich Pillichshammer

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Abstract

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Article information

Source
Funct. Approx. Comment. Math., Volume 58, Number 1 (2018), 89-104.

Dates
First available in Project Euclid: 5 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1493949628

Digital Object Identifier
doi:10.7169/facm/1649

Mathematical Reviews number (MathSciNet)
MR3780036

Zentralblatt MATH identifier
06924918

Subjects
Primary: 41A25: Rate of convergence, degree of approximation
Secondary: 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 65D15: Algorithms for functional approximation 65Y20: Complexity and performance of numerical algorithms [See also 68Q25]

Keywords
multivariate approximation Walsh space Korobov space hybrid function space

Citation

Kritzer, Peter; Laimer, Helene; Pillichshammer, Friedrich. Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces. Funct. Approx. Comment. Math. 58 (2018), no. 1, 89--104. doi:10.7169/facm/1649. https://projecteuclid.org/euclid.facm/1493949628


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