# Convolution on group with measure

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: $a (f * g) = (a f) * g = f * (a g) \,$

1. Time invariant: $f(t+\tau)*g(t)=h(t+\tau), \forall t, \tau$ in the group.

2. 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.

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 $(f * g)(x) = \int_G f(y) g(-y+x)\,d\lambda(y). \,$

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I am not sure what your definitions are here. What are f, g, h supposed to be? – Qiaochu Yuan Nov 25 '10 at 0:33
Lebesgue-integrable real-valued functions defined on a group equipped with measure – Tim Nov 25 '10 at 0:35
What kind of measure? Why are you adding real numbers to t in property 4 if t is supposed to be an element of some other group? – Qiaochu Yuan Nov 25 '10 at 0:36
Sorry, I made mistake, $\tau$ is in the group. The measure now is a general one. If some property does not hold, you can put more restriction or property on the measure, the group or the functions. – Tim Nov 25 '10 at 0:41
Generally you want the measure to be translation-invariant from at least one side (ideally both), i.e. a Haar measure, and the group to be locally compact and $\sigma$-compact. Most of these properties are easy exercises to verify or disprove in such cases. For example, commutativity (1) holds iff the group is abelian. Why not try them before asking? – Nate Eldredge Nov 25 '10 at 5:19

The proofs of all these assertions in the case when your group is $\mathbb R$, can be found in any elementary book defining convolution. These same proofs will work in the case of arbitrary groups.