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

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

when this quantity exists.

Results

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

$\begin{array}{cc}s{\ast }_{l}r(t)& ={\int }_0^{\mathrm{\infty }}s\left(\frac{t}{a}\right)r(a)\frac{da}{a}\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}s\left(\frac{t}{e^u}\right)r(e^u)du\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}s\left(e^{\log t-u}\right)r(e^u)du.\end{array}$

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

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

See also

• Mellin transform