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=(xm,n)\documentclass[12pt]{minimal}
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\begin{document}$x=(x_{m,n})$\end{document} is a double sequence that is mapped into ℓ2\documentclass[12pt]{minimal}
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\begin{document}$\ell_{2}$\end{document} by the four-dimensional Borel method and the double sequence x satisfies ∑m=0∞∑n=0∞|Δ10xm,n|mn<∞\documentclass[12pt]{minimal}
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\begin{document}$\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{10} x_{m,n}|\sqrt {mn}<\infty$\end{document} and ∑m=0∞∑n=0∞|Δ01xm,n|mn<∞\documentclass[12pt]{minimal}
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\begin{document}$\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{01} x_{m,n}|\sqrt {mn}<\infty$\end{document}, then x itself is in ℓ2\documentclass[12pt]{minimal}
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\begin{document}$\ell_{2}$\end{document}.
