# What is a rigorous statement for “linear time-invariant systems can be represented as convolutions”?

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'"

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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).

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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.

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As a more "down to earth" answer, I would say that linear systems have linear solutions, and convolution is a linear operator (or possibly bi-linear, based on the type of convolution) and as such the solutions of these equations can be represented as convolutions. The time-invariant property probably imposes some additional restrictions on the properties of the convolution.

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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 f\in 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.

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