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A Note on Some Prime Hausdorff Methods of Summability

Published online by Cambridge University Press:  20 November 2018

M. R. Parameswaran
M. R. Parameswaran*
Affiliation:
Dept. of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada
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Given a matrix A = (ank) (n, k = 0, 1, 2, …), let (A) denote the set of all sequences x = {xk} such that {An(x)} ∊ c where An(x) = Σk=0∞ankxk (n≥0) and c denotes the set of all convergent sequences. It is well known (see e.g. Zeller [6] or Zeller and Beekmann [7], p. 48) that given an unbounded sequence x, there exists a regular (=permanent) matrix A with ank = 0 for k > n (and indeed with ann ≠ 0) such that (A) = c ⊕x, the linear space spanned by c and x. We call A an Einfolgenverfahren. (See [7].) In [4] Rhoades considered, inconclusively, the question whether there exists a Hausdorff matrix H such that (H)= c ⊕ x (for arbitrary unbounded sequence x).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 2 , June 1975 , pp. 299 - 301
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
2. Jakimovski, A. and Parameswaran, M. R., Generalized Tauberian theorems for summability- (A), Quart. J. Math. Oxford (2) 9 (1958), 290-298.Google Scholar
3. Parameswaran, M. R., Remark on the structure of the summability field of a Hausdorff matrix, Proc. Nat. Inst. Sci. India (Sec. A) 27 (1961), 175-177.Google Scholar
4. Rhoades, B. E., Some structural properties of Hausdorjf matrices, Bull. Amer. Math. Soc. 65(1959), 9-11.Google Scholar
5. Rhoades, B. E., Size of convergence domains for known Hausdorjf prime matrices, J. Math. Analysis and Appl. 19 (1967), 457-468.Google Scholar
6. Zeller, K., Merkwiirdigkeiten bei Matrixverfahren; Einfolgenverfahren, Arch. Math. 4 (1953), 1-5.Google Scholar
7. Zeller, K. and Beekman, W., Theorie der Limitierungsverfahren (Berlin 1958/1970).Google Scholar