In this paper we study sequence spaces that arise from the concept of strong
weighted mean summability. Let q = (qn)
be a sequence of positive terms and set
Qn =
[sum ]nk=1qk.
Then the weighted mean matrix Mq =
(ank) is defined by
formula here
It is well known that Mq defines a regular summability method
if and only if Qn→∞. Passing to strong
summability, we let 0<p<∞. Then
formula here
are the spaces of all sequences that are strongly
Mq-summable with index p to 0,
strongly Mq-summable with index p and strongly
Mq-bounded with index p, respectively.
The most important special case is obtained by taking
Mq = C1, the Cesàro matrix,
which leads to the familiar sequence spaces
formula here
respectively, see [4, 21]. We remark that
strong summability was first studied by
Hardy and Littlewood [8] in 1913 when they applied
strong Cesàro summability of
index 1 and 2 to Fourier series; orthogonal series have remained the main area of
application for strong summability. See [32, §6] for further references.
When we abstract from the needs of summability theory certain features of the
above sequence spaces become irrelevant; for instance, the qk simply constitute a
diagonal transform. Hence, from a sequence space theoretic point of view we are led
to study the spaces