Some Tauberian theorems for four-dimensional Euler and Borel summability

Abstract

The four-dimensional summability methods of Euler and Borel are studied as mappings from absolutely convergent double sequences into themselves. Also the following Tauberian results are proved: if $x=(x_{m,n})$ is a double sequence that is mapped into $\ell_{2}$ by the four-dimensional Borel method and the double sequence x satisfies $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{10} x_{m,n}|\sqrt {mn}<\infty$ and $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{01} x_{m,n}|\sqrt {mn}<\infty$ , then x itself is in $\ell_{2}$ .