Sharp convergence bounds for sums of POD and SPOD weights

TL;DR

This paper derives sharp convergence bounds for sums involving POD and SPOD weights, characterizing their finiteness and growth behavior, which advances understanding of their mathematical properties in high-dimensional settings.

Contribution

It provides necessary and sufficient conditions for the convergence of sums with POD weights and characterizes their asymptotic growth, extending results to SPOD weights.

Findings

Convergence of sums depends on the summability of the sequence b4_j.

Growth of sums is asymptotically proportional to b1^{1/( ho-\sigma)}.

Results generalize to smoothness-driven weights, broadening applicability.

Abstract

This work analyzes the convergence of sums of the form $S_{\boldsymbol{\gamma}}(m)=\sum_{v\subseteq \mathbb{N}}\gamma_v m^{|v|}$, where $\gamma_v$ are product and order dependent (POD) weights. We establish that for nonnegative sequence $\{\Upsilon_j\mid j\in \mathbb{N}\}$, $$\sum_{v\subseteq \mathbb{N}} |v|! m^{|v|}\prod_{j\in v} \Upsilon_j<\infty \text{ for } m>0 \text{ if and only if } \sum_{j=1}^\infty \Upsilon_j<\infty.$$ We further characterize the growth of $S_{\boldsymbol{\gamma}}(m)$ when $\gamma_v=(|v|!)^{\sigma}\prod_{j\in v}j^{-\rho}$ and prove that $\log S_{\boldsymbol{\gamma}}(m)$ exhibits asymptotic order $m^{1/(\rho-\sigma)}$ when $\rho>\sigma$. All results are subsequently generalized to smoothness-driven product and order dependent (SPOD) weights.