An application of Banach limits
HTML articles powered by AMS MathViewer
- by Z. U. Ahmad and Mursaleen PDF
- Proc. Amer. Math. Soc. 103 (1988), 244-246 Request permission
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 $\left \| \mathcal {L} \right \| = 1$, 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
-
S. Banach, Theorie des operations lineaires, PWN, Warszawa, 1932.
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
- G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190. MR 27868, DOI 10.1007/BF02393648
- Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 133, 77–86. MR 688425, DOI 10.1093/qmath/34.1.77
- Ralph A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81–94. MR 154005
- Paul Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104–110. MR 306763, DOI 10.1090/S0002-9939-1972-0306763-0
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 244-246
- MSC: Primary 40C05; Secondary 46A45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938676-7
- MathSciNet review: 938676