Logarithmic convolution: Difference between revisions

Revision as of 07:24, 4 October 2016

Revision as of 16:10, 2 May 2021

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The logarithmic convolution can be related to the ordinary [[convolution]] by changing the [[Variable (mathematics)|variable]] from <math>t</math> to <math>v = \log t</math>:

: <math> \begin{align}
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 </math>\\

:<math>& = \int_{-\infty}^\infty s \left(e^{\log t - u}\right)r(e^u) \, du.</math>

\end{align}</math>

Define <math>f(v) = s(e^v)</math> and <math>g(v) = r(e^v)</math> and let <math>v = \log t</math>, then

:<math> s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).\, </math>

{{PlanetMath attribution|id=5995|title=logarithmic convolution}}