Logarithmic convolution

In mathematics, the scale convolution of two functions $s(t)$ and $r(t)$, also known as their logarithmic convolution is defined as the function
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$s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a}$

when this quantity exists.

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 Creative Commons Attribution/Share-Alike License.