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STRONG WEIGHTED MEAN SUMMABILITY AND KUTTNER'S THEOREM

Published online by Cambridge University Press:  01 June 1999

KARL-GOSWIN GROSSE-ERDMANN
KARL-GOSWIN GROSSE-ERDMANN
Affiliation:
Fachbereich Mathematik, Fernuniversität Hagen, Lützowstrasse 125, D-58084 Hagen, Germany
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Abstract

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

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 987 - 1002
Copyright
The London Mathematical Society 1999

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