One-sided Tauberian conditions and double sequences (Q1410431)

The linear transformation \[ \tau(x,y):= \sum^\infty_{k,\ell=0} c_{k\ell} (x,y) s_{k\ell} \] is considered for double sequences \(S=(s_{k\ell})\), where \(c_{k\ell} (x,y) \geq 0\) with \(k,\ell\in N_0\); \(x,y\in X\); and \(X\) is either \(N_0\) or \([0, \infty)\). A double sequence \(S\) is said to be \(C\)-summable to \(s\) if \[ \tau(x,y) \to s\quad \text{as}\quad x,y\to \infty\quad \text{in}\quad X; \] in short: \(C\)-\(\lim s_{mn} =s\). Furthermore, \(S\) is said to be boundedly \(C\)-summable to \(s\) if \[ C\text{-lim }s_{mn} =s\quad \text{and}\quad \sup_{x,y\in X} |\tau(x,y)|< \infty; \] in short: \(bC\)-lim \(s_{mn} = s\). Such a summability method \(C\) is said to be boundedly regular if for any sequence \(S\) that boundedly converges (in Pringsheim's sense) to \(s\), in short: \(b\)-\(\lim s_{mn} =s\), we have \(b C\)-\(\lim s_{mn} =s\). The characterization of bounded regularity of a summability method \(C\) is well known. The converse implication, from \(b C\)-\(\lim s_{mn} =s\) to \(b\)-\(\lim s_{mn} =s\), holds only under additional (so-called Tauberian) conditions on \(S\). In a joint paper with \textit{S. Baron} [J. Math. Anal. Appl. 211, 574-589 (1977; Zbl 0883.40002)], the present author established two-sided Tauberian conditions for a certain subclass of summability methods. In the present paper, one-sided Tauberian conditions are established.