fourier analysis - What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?

An abstract result in Banach algebra theory, known as Wendel's Theorem, tells us that the multiplier algebra of L^1(G) is M(G), the measure algebra, for any locally compact group G.

So, if G=R the reals, this says that if T:L^1(R) rightarrow L^1(R) is a bounded linear map which commutes with translations, then there is some measure mu on R such that T(f) = mu * f for all fin L^1(R). (And maybe this special case was known before Wendel?)

I don't know much about distributions, but this general area falls into the theory of "Multipliers" I believe.