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Compactness properties of weighted summation operators on trees

100%
Lifshits M., Linde W.
Studia Mathematica
|
2011
|
tom 202
|
nr 1
17-47
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.
2
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On the supremum of random Dirichlet polynomials

100%
Lifshits M., Weber M.
Studia Mathematica
|
2007
|
tom 182
|
nr 1
41-65
EN
We study the supremum of some random Dirichlet polynomials $D_{N}(t) = ∑_{n=2}^{N} εₙdₙn^{-σ-it}$, where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials $∑_{n∈ 𝓔_{τ}} εₙn^{-σ-it}$, $𝓔_{τ} = {2 ≤ n ≤ N : P⁺(n) ≤ p_{τ}}$, P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, $𝔼 sup_{t∈ ℝ} |∑_{n=2}^{N} εₙn^{-σ-it}| ≈ (N^{1-σ})/(log N)$. The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.
3
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Compactness properties of weighted summation operators on trees-the critical case

100%
Lifshits M., Linde W.
Studia Mathematica
|
2011
|
tom 206
|
nr 1
75-96
EN
The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill this gap. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed by the first-named author for a particular critical operator on the binary tree. Those problems appeared in a natural way during the study of compactness properties of certain Volterra integral operators in a critical case.
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