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Cesàro iterations of Hausdorff matrices

Published online by Cambridge University Press:  24 October 2008

B. Kuttner
B. Kuttner
Affiliation:
University of Birmingham
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Extract

1. Jurkat and Peyerimhoff(3) (or see (4), pages 50–56) have given a definition of a function, f(A), of an infinite matrix A = (ank). In particular, they considered Aα, where a is any real constant. This is defined wherever A is normal; that is to say, when ank = 0 for k > n, and when, for all n, ann ±0. They also defined the ‘Cesàro iteration’ Aα. This is obtained by defining a matrix B = (bnn) by

(so that every diagonal element of B is 1), forming Bα, and then dividing each row by the row sum; that is to say, if Bα = (b(α)nk), we take

where

The definition requires, of course, that ≠ 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 1 , January 1975 , pp. 109 - 118
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Borwein, D.Theorems on some methods of summability. Quart. J. of Math. Oxford Ser. 2 9 (1958), 310–316.CrossRefGoogle Scholar
(2)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(3)Jurkat, W. and Peyerimaoff, A.Tuber Äquivalenzprobleme und andere Limitierungs theoretische Fragen bei Halbgruppen positiva Matrizen. Math. Ann. 159 (1965), 234–251.CrossRefGoogle Scholar
(4)Peyerimhoff, A.Lectures on Summability. (Lecture Notes in Mathematics, 107; Springér Verlag, 1969).CrossRefGoogle Scholar