Logarithmic Convolution

In mathematics, the scale convolution of two functions $s(t)$ and $r(t)$, also known as their logarithmic convolution or log-volution is defined as the function

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

when this quantity exists.

Results

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

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

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

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

See also

* Mellin transform

References

External links

{{Use dmy dates, date=September 2024

Logarithms