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}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}

when this quantity exists.

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

	$\displaystyle s\ast_{l}r(t)$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}=\int_{-\infty}^{% \infty}s(\frac{t}{e^{u}})r(e^{u})du$
		$\displaystyle=$	$\displaystyle\int_{-\infty}^{\infty}s(e^{\log t-u})r(e^{u})du$

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).