## 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.)

### Keywords:

Walsh functions; sigma regular matrix
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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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