Logarithmic convolution: Difference between revisions

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In [[mathematics]], the '''scale convolution''' of two [[Function (mathematics)|functions]] <math>s(t)</math> and <math>r(t)</math>, also known as their '''logarithmic convolution''' or '''log-volution'''<ref>{{Cite book|title= An Introduction to Exotic Option Pricing | series = Chapman and Hall/CRC Financial Mathematics Series | author = Peter Buchen | publisher = CRC Press| date = 2012 | ISBN = 9781420091021}}</ref> is defined as the function<ref name=pm>{{Cite web|url=https://planetmath.org/logarithmicconvolution|work=Planet Math| title = logarithmic convolution |date=22 March 2013|access-date=15 September 2024}}</ref>

{{unreferenced|date=October 2010}}

The '''scale convolution''' of two functions <math>s(t)</math> and <math>r(t)</math>, also known as their '''logarithmic convolution''' is defined as the function<br>dodo

:<math> s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a}</math>

:<math> s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a}</math>

when this quantity exists.

==Results==

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>:<ref name=pm />

: <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>

==See also==

* [[Mellin transform]]

==References==

{{Reflist}}

==External links==

{{PlanetMath attribution|id=5995|title=logarithmic convolution|access-date=12 August 2006}}

{{Authority control}}

{{Use dmy dates|date=September 2024}}

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

[[Category:Logarithms]]