Compactness Properties of Weighted Summation Operators on Trees

TL;DR

This paper studies the compactness of weighted summation operators on trees, providing estimates for their entropy numbers and applying results to various tree structures and weights, with implications for Gaussian schemes.

Contribution

It introduces a new metric-based approach to analyze compactness of weighted summation operators on trees, with explicit entropy number estimates and applications to probabilistic models.

Findings

Derived two-sided estimates for entropy numbers of the operators.

Applied results to specific tree structures like binary and biased trees.

Connected compactness properties to probabilistic Gaussian summation schemes.

Abstract

We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$ (V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s)\,,\quad t\in T\;, $$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees.