In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of this theorem, or refer a book that includes it?
Edit: For example, would the following be a correct statement?
"Let S' be the space of tempered distributions. If L is a linear operator on S' that commutes with translations, then there exists a distribution h in S' such that Lf = f*h for all f in S'"
Best Answer
I think the result you are looking for is the following: Let T be a linear continuous and translation invariant operator mapping S into S' (rather than S' into S'). Then there exists a distribution K s.t. Tf = f*K, for every f in S.
The continuity of T is referred to the usual Frechet topology on S and the weak dual topology on S' (you want f -> to be a continuous linear functional on S for every g in S). You can find a proof (by Sobolev embedding) on Introduction to Fourier analysis on Euclidean spaces (E. Stein).
From this you can prove analogous results for L^p spaces by embedding (translation invariantly) S in L^p and L^q in S'.
To rephrase everything in the language of multipliers it suffices to remember that the Fourier transform F is a topological isomorphism of S' and that F(f*K) = F(f)F(K) whenever K is a tempered distribution and f is a Schwartz function. Than the operator T is a multiplier operator F(Tf) = m F(f) for m = F(K).