  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.
