Logarithmic convolution

The scale convolution of two functions ${\displaystyle s(t)}$ and ${\displaystyle r(t)}$, also known as their logarithmic convolution is defined as the function

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

when this quantity exists.

Results

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

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

${\displaystyle =\int _{-\infty }^{\infty }s\left(e^{\log t-u}\right)r(e^{u})du.}$ 

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

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

logarithmic convolution at PlanetMath.