Logarithmic convolution

In mathematics, the scale convolution of two functions ${\displaystyle s(t)}$ and ${\displaystyle r(t)}$, also known as their logarithmic convolution or log-volution^[1] is defined as the function^[2]

${\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}}$

when this quantity exists.