Logarithmic convolution

  In mathematics, the scale convolution of two functions ${\displaystyle s(t)}$ and ${\displaystyle r(t)}$, also known as their logarithmic convolution is defined as the function^[1]

${\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}}$

when this quantity exists.

Results

The logarithmic convolution can be related to the ordinary convolution by changing the variable from ${\displaystyle t}$  to ${\displaystyle v=\log t}$ :^[1]

${\displaystyle {\begin{aligned}s*_{l}r(t)&=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}\\&=\int _{-\infty }^{\infty }s\left({\frac {t}{e^{u}}}\right)r(e^{u})\,du\\&=\int _{-\infty }^{\infty }s\left(e^{\log t-u}\right)r(e^{u})\,du.\end{aligned}}}$ 

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

${\displaystyle s*_{l}r(v)=f*g(v)=g*f(v)=r*_{l}s(v).}$ 

References

1. "logarithmic convolution". Planet Math. 22 March 2013. Retrieved 15 September 2024.

External links

This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.