Sharp convergence bounds for sums of POD and SPOD weights

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.