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We investigate compactness properties of weighted summation operators $V_{α,σ}$ as mappings from ℓ₁(T) into $ℓ_{q}(T)$ for some q ∈ (1,∞). Those operators are defined by
$(V_{α,σ}x)(t) : = α(t) ∑_{s⪰t} σ(s)x(s)$, t ∈ T,
where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $eₙ(V_{α,σ})$, the (dyadic) entropy numbers of $V_{α,σ}$. The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t) decreasing either polynomially or exponentially. We also give some probabilistic applications to Gaussian summation schemes on trees.
$(V_{α,σ}x)(t) : = α(t) ∑_{s⪰t} σ(s)x(s)$, t ∈ T,
where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $eₙ(V_{α,σ})$, the (dyadic) entropy numbers of $V_{α,σ}$. The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t) decreasing either polynomially or exponentially. We also give some probabilistic applications to Gaussian summation schemes on trees.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
17-47
Opis fizyczny
Daty
wydano
2011
Twórcy
autor
- Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl. 2, 198504 Stary Peterhof, Russia
autor
- Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany
Bibliografia
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-1-2