Schwartz-Bruhat function

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In mathematics, a Schwartz-Bruhat function is a function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz-Bruhat functions.

[edit] Definitions

On a real vector space, the Schwartz-Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing).
On a torus, the Schwartz-Bruhat functions are the smooth functions.
On a sum of copies of the integers, the Schwartz-Bruhat functions are the rapidly decreasing functions.
On an elementary group (i.e. an abelian locally compact group that is a product of copies of the reals, the integers, the circle group, and finite groups), the Schwartz-Bruhat functions are the smooth functions all of whose derivatives are rapidly decreasing.
On a general locally compact abelian group G, let A be a compactly generated subgroup, and B a compact subgroup of A such that B/A is elementary. Then the pullback of a Schwartz-Bruhat function on B/A is a Schwartz-Bruhat function on G, and all Schwartz-Bruhat functions on G are obtained like this for suitable A and B. (The space of Schwartz-Bruhat functions on G is topologized with the inductive limit topology.)
In particular, on the ring of adeles over a number field or function field, the Schwartz-Bruhat functions are linear combinations of products of Schwartz functions on the infinite part and locally constant functions of compact support at the non-archimedean places (equal to the characteristic function of the integers at all but a finite number of places).

[edit] Properties

The Fourier transform of a Schwartz-Bruhat function on a locally compact abelian group is a Schwartz-Bruhat function on the Pontryagin dual group. Consequently the Fourier transform takes tempered distributions on such a group to tempered distributions on the dual group.

[edit] References

Osborne, M. Scott On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups. J. Functional Analysis 19 (1975), 40--49.