Suppose \(f,g\colon \mathbb{R}^d\to \mathbb{C}\). The convolution of \(f\) and \(g\) is defined by
\[ f\ast g(x):=\int_{\mathbb{R}^d} f(y)g(x-y) \,dy = \int_{\mathbb{R}^d} f(x-y)g(y) \,dy, \]if the integrals exist.
Remark
- The convolution of two functions can be interpreted as the weighted mean value of \(f\) where \(g\) is the weight, i.e. \(g\) defines a measure which is absolutely continuous with respect to the Lebesgue measure.
- The convolution fulfills different algebraic properties. If \(f,g,h\) are suitable functions and \(\alpha\) is some scalar, the following identities hold:
- commutativity: \(f\ast g=g\ast f\)
- associativity: \(f\ast (g\ast h)=(f\ast g)\ast h\)
- distributivity: \(f\ast (g+h)=f\ast h+f\ast g\)
- scalar associativity: \(\alpha(f\ast g)=\alpha f\ast g=f\ast \alpha g\)