I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain.
For convolution on Lebesgue-integrable real-valued functions defined on a group equipped with measure, does each of the following properties holds? If not, what extra requirements needed on the group, measure and/or functions in order that it can become true:
(1) Commutativity: $f * g = g * f \,$
(2) Associativity: $f * (g * h) = (f * g) * h \,$
(3) Linearity (This, I am more sure, holds, but I might be wrong):
Distributivity: $f * (g + h) = (f * g) + (f * h) \,$
Associativity with scalar multiplication: $ a (f * g) = (a f) * g = f * (a g) \, $
(4) Time invariant: $f(t+\tau)*g(t)=h(t+\tau), \forall t, \tau$ in the group.
(5) the reverse of (3) and (4), i.e.: Any linear and time-invariant operation on the set of Lebesgue-integrable real-valued functions must be the convolution with some Lebesgue-integrable real-valued function.
Thanks and regards!
The definition for the convolution on a group with measure is defined according to Wikipedia:
If G is a suitable group endowed with a measure λ, and if f and g are real or complex valued integrable functions on G, then we can define their convolution by
$(f * g)(x) = \int_G f(y) g(-y+x)\,d\lambda(y). \,$