Tauberian theorems for Abel limitability method
Suppose \(u=(u_n)\) is a sequence of real numbers and suppose that as \(n\to\infty\) we have \(u_n\to A(u)\). In this case \(u\) is also Abel summable to \(A(u)\), i.e., as \(x\to1\), we have \((1-x)\sum_{n=0}^{\infty}u_{n}x^n=A(u)\). The authors formulate new Tauberian conditions to prove converse statements. The theorems are formulated in terms of the operators \(I,S\) and \(T\) defined by \((Iu)_n=u_n\), \((Su)_n=\sum_{i=0}^n u_{i}/(n+1)\) and \((Tu)_n=n(u_n-u_{n-1})\) and generalise results of \textit{A. Tauber} [Monatsh. f. Math. 8, 273--277 (1897; JFM 28.0221.02)]. In the main result of the paper the authors prove that if for some integer \(m\geq1\) the sequence \((I-S)^{m+1}u\) is slowly oscillating, then Abel summability of \(u\) to \(A(u)\) implies that \(u_n\to A(u)\).
scientific article; zbMATH DE number 5292963