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Almost convergence of double sequences and strong regularity of summability matrices

Published online by Cambridge University Press:  24 October 2008

F. Móricz  and
B. E. Rhoades
F. Móricz
Affiliation:
University of Szeged, Bolyai Institute, Aradi vertanuk tere 1, 6720 Szeged, Hungary
B. E. Rhoades
Affiliation:
Indiana University, Department of Mathematics, Bloomington, Indiana 47405, U.S.A.
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Extract

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

that is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 283 - 294
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Lorentz, G. G.. A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167–190.CrossRefGoogle Scholar
[2]Rhoades, B. E.. Some applications of strong regularity to Markov chains and fixed point theorems. In Approximation Theory, vol. III (Academic Press, 1980), pp. 735–740.Google Scholar
[3]Rhoades, B. E. and Shi, Xianliang. Another way to characterize strong regularity. In Approximation Theory and Applications (Pitman, 1985), pp. 173–176.Google Scholar
[4]Robinson, G. M.. Divergent double sequences and series. Trans. Amer. Math. Soc. 28 (1926), 50–73.CrossRefGoogle Scholar