Convolution invariant linear functionals and applications to summability methods

We study topologically invariant means on $L^{\mathrm{\infty }}(\mathbb{R})$, the set of all essentially bounded functions on the real line, and prove that invariance with respect to a single convolution operator is sufficient for a mean to be topologically invariant. We also consider some applications of this result to summability methods. In particular, the notion of almost convergence is introduced for a function in $L^{\mathrm{\infty }}(\mathbb{R})$, and a Tauberian theorem concerning almost convergence and a summability method defined by a Wiener kernel is obtained. Further, for the $C_{\mathrm{\infty }}$ summability method, which is defined by the limit of H"{o}lder summability methods, we provide a necessary and sufficient condition for a given function to be $C_{\mathrm{\infty }}$ summable.