Logarithmic convolution

  In mathematics, the scale convolution of two functions $s(t)$ and $r(t)$, also known as their logarithmic convolution or log-volution is defined as the function

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

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

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

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

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