PlanetMath: logarithmic convolution

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logarithmic convolution

(Definition)

The scale convolution of two functions and , also known as their logarithmic convolution is defined as the function

$\displaystyle s \ast_l r(t) = r \ast_l s(t) = \int_0^\infty s(\frac{t}{a})r(a) \frac{da}{a}$

when this quantity exists.

The logarithmic convolution can be related to the ordinary convolution by changing the variable from 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

and

and let $v = \log t$ , then

$\displaystyle s \ast_l r(v) = f \ast g(v) = g \ast f(v) = r \ast_l s(v).$

"logarithmic convolution" is owned by swiftset.

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