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 $v=\log t$ :

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).\,

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

gollark: ```lualocal func_mt = {}debug.setmetatable(function() end, func_mt)function func_mt.__sub(lhs, rhs) return function(...) return lhs(rhs(...)) endendfunction func_mt.__add(lhs, rhs) return function(...) return rhs(lhs(...)) endend```

gollark: No clue what `..`, unary `-`, `/`, `*` or `^` should do.

gollark: `+` is right-to-left compose, `-` is left-to-right compose.

gollark: We could add that as their tostring method instead.

gollark: What should we do with functions? Composition?

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