In mathematics, the scale convolution of two functions s(t) and r(t), also known as their logarithmic convolution is defined as the function
\( \int_0^\infty s\left(\frac{t}{a}\right)r(a) \, \frac{da}{a} \)
when this quantity exists.
Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from t to \( v = \log t \):
\( {\displaystyle {\begin{aligned}s*_{l}r(t)&=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}\\&=\int _{-\infty }^{\infty }s\left({\frac {t}{e^{u}}}\right)r(e^{u})\,du\\&=\int _{-\infty }^{\infty }s\left(e^{\log t-u}\right)r(e^{u})\,du.\end{aligned}}} \)
Define \( f(v) = s(e^v) \) and \( g(v) = r(e^v) \) and let \( v = \log t \), then
\( {\displaystyle s*_{l}r(v)=f*g(v)=g*f(v)=r*_{l}s(v).} \)
This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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