Logarithmic convolution

In mathematics, the scale convolution of two functions $s(t)$ and $r(t)$ , also known as their logarithmic convolution or log-volution{{Cite book|title= An Introduction to Exotic Option Pricing | series = Chapman and Hall/CRC Financial Mathematics Series | author = Peter Buchen | publisher = CRC Press| date = 2012 | ISBN = 9781420091021}} is defined as the function{{Cite web|url=https://planetmath.org/logarithmicconvolution|work=Planet Math| title = logarithmic convolution |date=22 March 2013|access-date=15 September 2024}}

: $s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a}$

when this quantity exists.

Results

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

: $\begin{align}
s *_l r(t) & = \int_0^\infty s \left(\frac{t}{a}\right)r(a) \, \frac{da}{a} \\
& =
\int_{-\infty}^\infty s\left(\frac{t}{e^u}\right) r(e^u) \, du \\
& = \int_{-\infty}^\infty s \left(e^{\log t - u}\right)r(e^u) \, du.
\end{align}$

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

: $s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).$

References

External links

{{PlanetMath attribution|id=5995|title=logarithmic convolution|access-date=12 August 2006}}

Category:Logarithms

Logarithmic convolution

Results

See also

References

External links