Logarithmic convolution

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

:$s\; *\_l\; r(t)\; =\; r\; *\_l\; s(t)\; =\; int\_0^infty\; sleft(frac\{t\}\{a\}\; ight)r(a)\; frac\{da\}\{a\}$

when this quantity exists.

Results

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

: $s\; *\_l\; r(t)\; =\; int\_0^infty\; sleft(frac\{t\}\{a\}\; ight)r(a)\; frac\{da\}\{a\}\; =int\_\{-infty\}^infty\; sleft(frac\{t\}\{e^u\}\; ight)\; r(e^u)\; du$

:$=\; int\_\{-infty\}^infty\; sleft(e^\{log\; t\; -\; u\}\; ight)r(e^u)\; du.$

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

:$s\; *\_l\; r(v)\; =\; f\; *\; g(v)\; =\; g\; *\; f(v)\; =\; r\; *\_l\; s(v).,$