Summability methods and Fourier analysis on ${\mathbb{R}}^{\times }$

Abstract

We introduce a certain class of summability methods on $L^{\infty }([1,\infty ))$ which is defined by convolution in the group algebra $L^1({\mathbb{R}}^{\times })$. This class contains an integral version of Cesàro summability method and we particularly give a necessary and sufficient condition for a summability method in the class to equivalent to this one. Also, we consider another class of summability methods concerned with convolution in the group algebra $L^1(\mathbb{R})$ and give similar results.

12 pages