Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense.

Móricz, Ferenc; Stadtmüller, U.

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A Tauberian theorem for the Ingham summation method

Vytas Zacharovas (2011)

Acta Arithmetica

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Tauberian theorems for Cesàro summable double integrals over $ℝ_{+}^{2}$

Ferenc Móricz (2000)

Studia Mathematica

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Given ⨍ ∈ $L_{l}^{1} o c (ℝ_{+}^{2})$ , denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these...