logarithmic convolution

Definition

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

$$s{\ast }_{l}r(t)=r{\ast }_{l}s(t)={\int }_0^{\mathrm{\infty }}s(\frac{t}{a})r(a)\frac{da}{a}$$

when this quantity exists.

Results

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

$s{\ast }_{l}r(t)$$=$${\displaystyle {\int }_0^{\mathrm{\infty }}}s({\displaystyle \frac{t}{a}})r(a){\displaystyle \frac{da}{a}}={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}s({\displaystyle \frac{t}{e^u}})r(e^u)𝑑u$

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

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

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