Abstract. In the paper we deal with strong summability in families of summability methods which depend on a continuous parameter and where two different methods are connected either by a Cesàro-type or Euler--Knopp-type or Riesz-type method. We prove various results for strong summability based on these families. As particular cases, the families of generalized Nörlund methods both in matrix and in integral form, the families of Cesàro, Euler--Knopp and Riesz methods and the family of Borel-type methods are considered. This paper extends the authors investigations on ordinary summability in the families mentioned above started in [21] and [22]. Our interest in strong summability was supported by different recent papers on strong summability of orthogonal, in particular Fourier series (e.g. [18], [13], [15]).

AMS Subject Classification (1991): 40F05, 40G05

Received February 12, 2003. (Registered under 5836/2009.)