Logarithmic convolution

In mathematics, the scale convolution of two functions $$s(t)$$ and $$r(t)$$, also known as their logarithmic convolution is defined as the function


 * $$ s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a}$$

when this quantity exists.

Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from $$t$$ to $$v = \log t$$:


 * $$\begin{align}

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

Define $$f(v) = s(e^v)$$ and $$g(v) = r(e^v)$$ and let $$v = \log t$$, then


 * $$ s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v). $$