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An application of Banach limits. (English) Zbl 0652.40009

Let \(\ell_{\infty}\) and c denote the Banach space of bounded and convergent sequences respectively, \(\sigma\) an injection of the set of positive integers into itself having no finite orbits, and T the operator defined on \(\ell_{\infty}\) by \(Ty(n)=y(\sigma n).\) A positive linear functional \({\mathcal L}\) with \(\| {\mathcal L}\| =1\) is called a \(\sigma\)- mean if \({\mathcal L}(y)={\mathcal L}(Ty)\) for all y in \(\ell_{\infty}\). A sequence y is said to be \(\sigma\)-convergent denoted \(y\in V\sigma\), if \({\mathcal L}(y)\) takes the same value, called \(\sigma\)-lim y, for all \(\sigma\)-means \({\mathcal L}\). P. Schaefer gave necessary and sufficient conditions on a matrix A to ensure that \(A(c)\subset V_{\sigma}\), and additional conditions ensuring that \(\sigma -\lim Ay=\lim y\) for all \(y\in c\). We call such matrices \(\sigma\)-regular. The authors prove one theorem using \(\sigma\)-regular matrices to sum the sequence of Walsh functions.
Reviewer: I.Sukla

MSC:

40C05 Matrix methods for summability
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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References:

[1] S. Banach, Theorie des operations lineaires, PWN, Warszawa, 1932.
[2] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372 – 414. · Zbl 0036.03604
[3] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167 – 190. · Zbl 0031.29501 · doi:10.1007/BF02393648
[4] Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 133, 77 – 86. · Zbl 0539.40006 · doi:10.1093/qmath/34.1.77
[5] Ralph A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81 – 94. · Zbl 0125.03201
[6] Paul Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104 – 110. · Zbl 0255.40003
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