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>

==Definition==

:<math> s *</math>₁<math>_l r(t) = r *</math>₁<math>_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a}</math>

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>

<math> s *</math>₁<math> r(t) = r *</math>₁<math> s(t) = \int_0^\infty s(\frac{t}{a})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}

<math> s *</math>₁<math> r(t) = \int_0^\infty s(\frac{t}{a})r(a) \frac{da}{a} =

s *_l r(t) & = \int_{-\infty}int_0^\infty s \left(\frac{t}{e^ua}\right) r(e^ua) du\, </math>\frac{da}{a} \\

& =

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

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

\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></math> s \ast_l r(v) = f \ast g(v) = g \ast f(v) = r \ast_l s(v). <math></math><br>

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

{{planetmath|id=5995|title=logarithmic convolution}}

[[Category:mathematics]]

==See also==

* [[Mellin transform]]

==References==

{{Reflist}}

==External links==

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

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

[[Category:mathematicsLogarithms]]