Logarithmic convolution: Difference between revisions

Revision as of 14:22, 15 September 2024

Revision as of 15:41, 15 September 2024

Line 3:

<!-- Once discussion is closed, please place on talk page: {{Old AfD multi|page=Logarithmic convolution|date=14 September 2024|result='''keep'''}} -->

<!-- End of AfD message, feel free to edit beyond this point --> 

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''' is defined as the function{{cn<ref name=pm>((Cite web|dateurl=Septemberhttps://planetmath.org/logarithmicconvolution|title = logarithmic 2024}}convolution

|date=22 March 2013|acess-date=15 September 2024}}</ref>

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

Line 10 ⟶ 11:

==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>:{{cn|date<ref name=Septemberpm 2024}}/>

: <math>\begin{align}