Schwartz–Bruhat function

In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, 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 A/B is elementary. Then the pullback of a Schwartz–Bruhat function on A/B 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 (1975). "On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups". J. Functional Analysis 19: 40–49. doi:10.1016/0022-1236(75)90005-1. 
• Gelfand, I. M.; et al. (1990). Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7.