Logarithmic convolution: Difference between revisions

Revision as of 12:42, 12 August 2006

Revision as of 20:10, 12 August 2006

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==Definition==

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>_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>

when this quantity exists.

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The logarithmic convolution can be related to the ordinary convolution by changing the variable from <math>t</math> to <math>v = \log t</math>:

: <math> s *</math>₁<math>_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>

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*_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}}