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{aligned}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{aligned}}

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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