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A Note on Some Prime Hausdorff Methods of Summability
Published online by Cambridge University Press: 20 November 2018
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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).
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- Copyright © Canadian Mathematical Society 1975
References
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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
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Zeller, K., Merkwiirdigkeiten bei Matrixverfahren; Einfolgenverfahren, Arch. Math.
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