For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication -
(f.g)(x) = f(x)g(x),
and convolution -
(f*g)(x) = ∫y+z=xf(y)g(z)
Both operations define commutative ring structures (possibly without identity) with the usual addition. (For that to make sense, we have to find a subset of functions that is closed under addition, multiplication, and convolution. If G is finite, this is not an issue, and if G is compact, we can consider infinitely differentiable functions, and if G is Rd, we can consider the Schwarz class of infinitely differentiable functions that decay at infinity faster than all polynomials, etc. As long as our class of functions doesn't satisfy any additional nontrivial algebraic identities, it doesn't matter what it is precisely.)
My question is simply: do these two commutative ring structures satisfy any additional nontrivial identities?
A "trivial" identity is just one that's a consequence of properties mentioned above: e. g., we have the identity
f*(g.h) = (h.g)*f,
but that follows from the fact that multiplication and convolution are separately commutative semigroup operations.
