THC Science

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).\,

@@ Line 1: / Line 1: @@
-==Definition==
 The '''scale convolution''' of two functions <math>s(t)</math> and <math>r(t)</math>, also known as their '''logarithmic convolution''' is defined as the function<br>
-<math> s *</math><sub>1</sub><math> r(t) = r *</math><sub>1</sub><math> s(t) = \int_0^\infty s(\frac{t}{a})r(a) \frac{da}{a}</math>
+:<math> s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a}</math>
 when this quantity exists.
@@ Line 10: / Line 8: @@
 The logarithmic convolution can be related to the ordinary convolution by changing the variable from <math>t</math> to <math>v = \log t</math>:
-<math> s *</math><sub>1</sub><math> r(t)  =  \int_0^\infty s(\frac{t}{a})r(a) \frac{da}{a} =
+: <math> s *_l r(t)  =  \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a} =
-\int_{-\infty}^\infty s(\frac{t}{e^u}) r(e^u) du </math>
+\int_{-\infty}^\infty s\left(\frac{t}{e^u}\right) r(e^u) du </math>
-:<math> =  \int_{-\infty}^\infty s(e^{\log t - u})r(e^u) du</math>
+:<math> =  \int_{-\infty}^\infty s\left(e^{\log t - u}\right)r(e^u) du.</math>
 Define <math>f(v) = s(e^v)</math> and <math>g(v) = r(e^v)</math> and let <math>v = \log t</math>, then
-<math></math> s \ast_l r(v) = f \ast g(v) = g \ast f(v) = r \ast_l s(v). <math></math><br>
+:<math> s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).\, </math>
 {{planetmath|id=5995|title=logarithmic convolution}}