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


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