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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
up vote 1 down vote accepted

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.

For instance, take the case of associativity. A proof for the case of real numbers is available at planetmath. The same proof technique carries over to the general case. All you need additionally is to formally state two theorems: 1. Fubini's theorem on integration wrt product measures, and 2. translation invariance of the measure.

Here note that you always have to use a Haar measure.

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