In 1945 Brudno presented the following important theorem: If
are regular summability matrix methods such that every bounded sequence summed by
is also summed by
, then it is summed by
to the same value. In 1960 Petersen extended Brudno’s theorem by using uniformly summable methods. The goal of this paper is to extend Petersen’s theorem to double sequences by using four dimensional matrix transformations and notion of uniformly summable methods for double sequences. In addition to this extension we shall also present an accessible analogue of this theorem.
In 1936 Hamilton presented a Silverman-Toeplitz type characterization of c″0 (i.e. the space of bounded double Pringsheim null sequences). In this paper we begin with the presentation of a notion of asymptotically statistical regular. Using this definition and the concept of maximum remaining difference for double sequence, we present the following Silverman-Toeplitz type characterization of double statistical rate of convergence: let A be a nonnegative c″0−c″0 summability matrix and let [x] and [y] be member of l″ such that
with [x] ∈ P0, and [y] ∈ Pδ for some δ > 0 then µ(Ax)
µ(Ay). In addition other implications and variations shall also be presented.
We introduce the concept of double uniform density of subsets of ℕ×ℕ and the study the corresponding convergence (namely, Iu-convergence) of double sequences. Further we solve an inequality related to the Iu-limit superior of bounded double sequences in line of .