Abstract. Hardy [2] characterized summability $(C, 1)$ of a single series of complex numbers by the ordinary convergence of another series. We extend this result to double series, while convergence in Pringsheim's sense should be replaced by regular convergence, introduced also by Hardy [1] and rediscovered by the first named author [3] (see also [5]). We also present an example showing that the assumption of regularity is indispensable for the validity of our extension.

AMS Subject Classification (1991): 40B05, 40G05

Keyword(s): Double series and sequences, convergence in Pringsheim's sense, regular convergence, (C, summability, 1, 1), (C, 1, 0), (C, and, 0, 1), regular summability, Cauchy convergence principle, double summation by parts

Received January 31, 1997. (Registered under 5778/2009.)