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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?

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"

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'"

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 version of this theorem?

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?

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"