Tauberian theorems for power series methods applied to double sequences
Let \(\{p_{mn}; m\geq 0, n\geq 0\}\) be a double sequence with \(p_{00}> 0\). Define \[ p(x,y)= \sum^\infty_{m=0} \sum^\infty_{n= 0} p_{mn}x^my^n\quad\text{for }x,y\in(0, 1),\qquad P_{mn}= \sum^m_{k=0} \sum^n_{\ell=0} p_{k\ell}, \] and assume \(P_{mn}\to\infty\) as \(m,n\to\infty\). For a complex double sequence \(\{s_{mn}\}\), define the weighted power series \[ \sigma(x,y)= \sum^\infty_{m=0} \sum^\infty_{n=0} {s_{mn}p_{mn}x^my^n\over p(x,y)} \] and the weighted mean \[ \sigma_{mn}= P^{-1}_{mn} \sum^m_{k=0} \sum^n_{\ell=0} p_{k\ell}s_{k\ell}. \] This paper explores connections between the convergence of \(\{s_{mn}\}\), \(\{\sigma_{mn}\}\) and \(\{\sigma(x,y)\}\), especially in the case where \(p_{mn}= p_mq_n\) for some nonnegative sequences \(\{p_m\}\) and \(\{q_n\}\). In this special case, under certain Tauberian conditions, \(s_{mn}\to s\) as \(m,n\to\infty\) if \(\{s_{mn}\}\) is summable to \(s\) in the power series mean (i.e., \(p(x,y)\sigma(x,y)\) converges for all \(x,y\in(0, 1)\) and \(\sigma(x,y)\to s\) as \(x,y\to1-\)); if, moreover, \(\sigma^*\equiv\sup\{\sigma(x,y): x,y\in(0, 1)\}<\infty\) then \(\sup_{m,n} |s_{mn}|< \infty\). Conditions are also given under which \(\sigma_{mn}\to s\) and \(\sup_{m,n} |\sigma_{mn}|< \infty\) whenever \(\sigma^*<\infty\) and \(s_{mn}\to s\) in the power series mean.
scientific article; zbMATH DE number 1053440
scientific article