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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An application of Banach limits

Authors: Z. U. Ahmad and Mursaleen
Journal: Proc. Amer. Math. Soc. 103 (1988), 244-246
MSC: Primary 40C05; Secondary 46A45
MathSciNet review: 938676
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Abstract: Let $ {l_\infty }$ denote the Banach space of bounded sequences, $ \sigma $ an injection of the set of positive integers into itself having no finite orbits, and $ T$ the operator defined on $ {l_\infty }$ by $ Ty\left( n \right) = y\left( {\sigma n} \right)$. A positive linear functional $ \mathcal{L}$ with $ \left\Vert \mathcal{L} \right\Vert = 1$, is called a $ \sigma $-mean if $ \mathcal{L}\left( y \right) = \mathcal{L}\left( {{T_y}} \right)$ for all $ y$ in $ {l_\infty }$. A sequence $ y$ is said to be $ \sigma $-convergent, denoted $ y \in {V_\sigma }$, if $ \mathcal{L}\left( y \right)$ takes the same value, called $ \sigma - \lim y$, for all $ \sigma $-means $ \mathcal{L}$. P. Schaefer [6] gave necessary and sufficient conditions on a matrix $ A$ to ensure that $ A\left( c \right) \subset {V_\sigma }$, where $ c$ is the space of convergent sequences, and additional conditions ensuring that $ \sigma - \lim Ay = \lim y$ for all $ y \in c$, denoting the class of matrices satisfying these conditions by $ {\left( {c,{V_\sigma }} \right)_1}$ and calling them the $ \sigma $-regular matrices. In this paper, we use such matrices to find the sum of a sequence of Walsh functions.

References [Enhancements On Off] (What's this?)

  • [1] S. Banach, Theorie des operations lineaires, PWN, Warszawa, 1932.
  • [2] N. J. Fine, On Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. MR 0032833 (11:352b)
  • [3] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190. MR 0027868 (10:367e)
  • [4] Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford (2) 34 (1983), 77-86. MR 688425 (84h:40004)
  • [5] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81-94. MR 0154005 (27:3965)
  • [6] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104-110. MR 0306763 (46:5885)

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Keywords: Walsh functions, Banach limits, infinite matrices
Article copyright: © Copyright 1988 American Mathematical Society

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