International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 40   | 1 | 2 |

## Section 1.3.2.5.3. Definition and examples of tempered distributions

G. Bricognea

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#### 1.3.2.5.3. Definition and examples of tempered distributions

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A distribution is said to be tempered if it can be extended into a continuous linear functional on .

If is the topological dual of , and if , then its restriction to is a tempered distribution; conversely, if is tempered, then its extension to is unique (because is dense in ), hence it defines an element S of . We may therefore identify and the space of tempered distributions.

A distribution with compact support is tempered, i.e. . By transposition of the corresponding properties of , it is readily established that the derivative, translate or product by a polynomial of a tempered distribution is still a tempered distribution.

These inclusion relations may be summarized as follows: since contains but is contained in , the reverse inclusions hold for the topological duals, and hence contains but is contained in .

A locally summable function f on will be said to be of polynomial growth if can be majorized by a polynomial in as . It is easily shown that such a function f defines a tempered distribution viaIn particular, polynomials over define tempered distributions, and so do functions in . The latter remark, together with the transposition identity (Section 1.3.2.4.4), invites the extension of and from to .