logarithmic convolution
Definition
The scale convolution of two functions s(t) and r(t), also known as their logarithmic convolution is defined as the function
s∗lr(t)=r∗ls(t)=∫∞0s(ta)r(a)daa |
when this quantity exists.
Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from t to v=logt:
s∗lr(t) | = | ∫∞0s(ta)r(a)daa=∫∞-∞s(teu)r(eu)𝑑u | ||
= | ∫∞-∞s(elogt-u)r(eu)𝑑u |
Define f(v)=s(ev) and g(v)=r(ev) and let v=logt, then
s∗lr(v)=f∗g(v)=g∗f(v)=r∗ls(v). |
Title | logarithmic convolution |
---|---|
Canonical name | LogarithmicConvolution |
Date of creation | 2013-03-22 14:28:26 |
Last modified on | 2013-03-22 14:28:26 |
Owner | swiftset (1337) |
Last modified by | swiftset (1337) |
Numerical id | 4 |
Author | swiftset (1337) |
Entry type | Definition |
Classification | msc 44A35 |
Synonym | scale convolution |
Related topic | Convolution |