Logarithmic convolution

In mathematics, the scale convolution of two functions [math]\displaystyle{ s(t) }[/math] and [math]\displaystyle{ r(t) }[/math], also known as their logarithmic convolution is defined as the function

[math]\displaystyle{ 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]\displaystyle{ t }[/math] to [math]\displaystyle{ v = \log t }[/math]:

[math]\displaystyle{ \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} }[/math]

Define [math]\displaystyle{ f(v) = s(e^v) }[/math] and [math]\displaystyle{ g(v) = r(e^v) }[/math] and let [math]\displaystyle{ v = \log t }[/math], then

[math]\displaystyle{ s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v). }[/math]