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-volution¹ is defined as the function²

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

when this quantity exists.

The logarithmic convolution can be related to the ordinary convolution by changing the variable from ${\displaystyle t}$ to ${\displaystyle v=\log t}$:²

${\displaystyle {\begin{aligned}s*_{l}r(t)&=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}\\&=\int _{-\infty }^{\infty }s\left({\frac {t}{e^{u}}}\right)r(e^{u})\,du\\&=\int _{-\infty }^{\infty }s\left(e^{\log t-u}\right)r(e^{u})\,du.\end{aligned}}}$

Define ${\displaystyle f(v)=s(e^{v})}$ and ${\displaystyle g(v)=r(e^{v})}$ and let ${\displaystyle v=\log t}$, then

${\displaystyle s*_{l}r(v)=f*g(v)=g*f(v)=r*_{l}s(v).}$