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

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 Ces\`{a}ro 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.