A Tauberian theorem for multiple power series (Q2812518)

Let \((a(i),i\in\mathbb{Z}_{+}^{n})\) denote a sequence of nonnegative real numbers with generating function \(A(s_{1},\dots,s_{n})\) and let \(m(k)=(m_{1}(k),\dots,m_{n}(k))\) denote a sequence of positive numbers such that \(\min_{i}m_{i}(k)\rightarrow \infty \) as \(k\rightarrow \infty \). The author considers the asymptotic behaviour of \(A(e^{-\lambda _{1}/m_{1}},\dots,e^{-\lambda _{n}/m_{n}})\) as \(k\rightarrow \infty \) and relates it to the asymptotic behaviour of the sequence \(a(\left[ x_{1}m_{1}\right] ,\dots,\left[ x_{n}m_{n}\right] )\) as \(k\rightarrow \infty\). Assuming that \((a(i))\) is one-sided weakly oscillating, the author proves a genuine Tauberian theorem together with some extensions. The results should be compared to [\textit{U. Stadtmüller} and \textit{R. Trautner}, J. Reine Angew. Math. 323, 127--138 (1981; Zbl 0445.44004)].