Logarithmic convolution

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

$s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a}$

when this quantity exists.

[edit] Results

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

$s *_l r(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a} = \int_{-\infty}^\infty s\left(\frac{t}{e^u}\right) r(e^u) du$

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

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

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

This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the GFDL.