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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

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