Schwartz-Bruhat function

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.

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).

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.

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.