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 *_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 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
:<math> s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).\, </math>