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• # Artykuł - szczegóły

## Studia Mathematica

2011 | 202 | 1 | 17-47

## Compactness properties of weighted summation operators on trees

EN

### Abstrakty

EN
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.

17-47

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