Logarithmic convolution: Difference between revisions

Revision as of 03:48, 13 September 2012

Revision as of 04:00, 13 October 2012

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{{Multiple issues|wikify = September 2012|dead end = September 2012|unreferenced = October 2010}}

TheIn mathematics, 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>

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 *_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