# logarithmic convolution

## Definition

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

 $s\ast_{l}r(t)=r\ast_{l}s(t)=\int_{0}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}$

when this quantity exists.

## Results

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

 $\displaystyle s\ast_{l}r(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}=\int_{-\infty}^{% \infty}s(\frac{t}{e^{u}})r(e^{u})du$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}s(e^{\log t-u})r(e^{u})du$

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

 $s\ast_{l}r(v)=f\ast g(v)=g\ast f(v)=r\ast_{l}s(v).$
Title logarithmic convolution LogarithmicConvolution 2013-03-22 14:28:26 2013-03-22 14:28:26 swiftset (1337) swiftset (1337) 4 swiftset (1337) Definition msc 44A35 scale convolution Convolution