An application of Banach limits

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.