On Strong Summability

Strict inclusion between strong and ordinary methods of summability.

“…The matrix P is regul r if and only if, in addition to (2) and (3), it satisfies (4) lim ΣΡη,,^Ι* n->oo v=Q Throughout this paper A is a positive number. As in [2] we define the strong summability method [P, @] A s follows.…”

“…In particular the conclusions hold if λ > μ > 0 and P is regul r, The summability method PQ referred to above, is defined by '{$"} is P()-convergent to s if {a n } s given by (1), is P-convergent to s.' Both Theorem 2 (i) and its converse are established in [4] for the case Q = L It is our principal purpose to investigate general conditions on the matrix P under which the inclusion relations in Theorems l and 2 are strict. In the next section we give some basic properties of strong summability and strong boundedness which will beuseful in our investigation.…”

Lacunary almost convergence and some new sequence spaces

“…Also, the strongly almost summable sequence was studied by Nanda [15]. The strongly summable sequences have been systematically investigated by Hamilton and Hill [7], Kuttner [10] and some others. The goal of this paper is to study the spaces of strongly lacunary almost summable sequences, which naturally come up for investigation and which will fill up a gap in the existing literature.…”

On Convergence with respect to an Ideal and a Family of Matrices

“…In particular, strong --summability and -statistical convergence are equivalent on bounded sequences (see also [7,Theorem 8]). More information on strong matrix summability can be found in [8] (for the case = 1) or [9].…”