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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.

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