Logarithmic convolution explained

s(t) and r(t) , also known as their logarithmic convolution is defined as the functions * _{l} r(t) = r * _{l} s(t) = \int ^{inf ty} _{0} s\left( (t)/(a) \right)r(a) (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 ^{inf ty} _{0} s \left( (t)/(a) \right)r(a) (da)/(a) \\ & = \int ^{inf ty} _{-inf ty} s\left( (t)/(e ^{u}) \right) r(e ^{u)} du \\ & = \int ^{inf ty} _{-inf ty} s \left(e ^{log} \right)r(e ^{u)} du. \end{align}

Define

f(v) = s(e ^{v)} and g(v) = r(e ^{v)} and let v = log t , thens * _{l} r(v) = f * g(v) = g * f(v) = r * _{l} s(v).