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

We introduce a certain class of summability methods on $L^{\infty}([1, \infty))$whichisdefinedbyconvolutioninthegroupalgebra$L^1($\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.