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
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 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:

 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.

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).\,

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