Compactness properties of weighted summation operators on trees

Mikhail Lifshits ; Werner Linde

Studia Mathematica, Tome 204 (2011), p. 17-47 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2011-01-01

Zbl 1232.47020

EUDML-ID : urn:eudml:doc:285501