This paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

Let A, B denote sequence-to-sequence matrix methods of summability and A · B the “dot” or iteration product defined by (A · B)x = A(Bx) for all sequences x for which this exists. Some inclusion relations are given involving the methods A, B, A · B, B · A and the method defined by the matrix product AB. We take A, B to be of certain types whose products have not been studied extensively before, e.g. H* · Ck or Ck · H* where H* is quasi-Hausdorff (and hence upper triangular) and Ck is a Cesàro matrix (which is lower triangular). The investigations show also a link between the “Product Property” A ⊂ A · B and the translativity properties of A and B.

The behaviour of summability transforms of power series outside their circles of convergence has been studied by many authors. In the case of the geometric series Luh [6] and Tomm [10] showed that there exist regular methods A which provide an analytic continuation into any given simply connected region G that contains the unit disc but not the point 1. Moreover, the Atransforms of the geometric series may be required to converge to any chosen analytic function on prescribed regions outside the unit circle. In this paper, these results are extended to power series representing other meromorphic functions. It is also shown that the summability methods involved may be chosen to be generalized weighted means previously introduced by Faulstich [1].