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Register a new userApproximation of functions with small mixed smoothness in the uniform norm
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Added by
Tino Ullrich
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.
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We study the embedding $text{id}: ell_p^b(ell_q^d) to ell_r^b(ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(text{id})$ provided that $0<p<rleq infty$ and $0<qleq uleq infty$. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed smoothness which settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.
The paper gives a constructive method, based on greedy algorithms, that provides for the classes of functions with small mixed smoothness the best possible in the sense of order approximation error for the $m$-term approximation with respect to the trigonometric system.
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In this paper we give exact values of the best $n$-term approximation widths of diagonal operators between $ell_p(mathbb{N})$ and $ell_q(mathbb{N})$ with $0<p,qleq infty$. The result will be applied to obtain the asymptotic constants of best $n$-term approximation widths of embeddings of function spaces with mixed smoothness by trigonometric system.
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weavers conjecture we mostly lose a $sqrt{log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.
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