TITLE: What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
QUESTION [7 upvotes]: 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'"

REPLY [2 votes]: The Schwartz kernel theorem seams relevant here. You might recall from your signal processing books that in the linear but non-time-invariant case we still get the output as a integral $\int K(x,y) f(y)dy$ where $f$ is the input. The kernel theorem makes this rigorous as I recall where, $K$ then can be a distribution. 
Once you have that theorem it is probably easy to get the statement you want.