Schwartz–Bruhat function
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued 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) and form the space . 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 , let  be a compactly generated subgroup, and  a compact subgroup of  such that  is elementary. Then the pullback of a Schwartz–Bruhat function on  is a Schwartz–Bruhat function on , and all Schwartz–Bruhat functions on  are obtained like this for suitable  and . (The space of Schwartz–Bruhat functions on  is endowed with the inductive limit topology.) On a non-archimedean local field , a Schwartz–Bruhat function is a locally constant function of compact support. In particular, on the ring of adeles  over a global field , the Schwartz–Bruhat functions  are finite linear combinations of the products  over each place  of , where each  is a Schwartz–Bruhat function on a local field  and  is the characteristic function on the ring of integers  for all but finitely many . (For the archimedean places of , the  are just the usual Schwartz functions on , while for the non-archimedean places the  are the Schwartz–Bruhat functions of non-archimedean local fields.) The space of Schwartz–Bruhat functions on the adeles  is defined to be the restricted tensor product  of Schwartz–Bruhat spaces  of local fields, where  is a finite set of places of . The elements of this space are of the form , where  for all  and  for all but finitely many . For each  we can write , which is finite and thus is well defined. Examples
Every Schwartz–Bruhat function  can be written as , where each , , and . This can be seen by observing that  being a local field implies that  by definition has compact support, i.e.,  has a finite subcover. Since every open set in  can be expressed as a disjoint union of open balls of the form  (for some   and ) we have

 . The function  must also be locally constant, so  for some . (As for  evaluated at zero,  is always included as a term.) On the rational adeles  all functions in the Schwartz–Bruhat space  are finite linear combinations of  over all rational primes , where , , and  for all but finitely many . The sets  and  are the field of p-adic numbers and ring of p-adic integers respectively. 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. Given the (additive) Haar measure on  the Schwartz–Bruhat space  is dense in the space

Applications
In algebraic number theory, the Schwartz–Bruhat functions on the adeles can be used to give an adelic version of the Poisson summation formula from analysis, i.e., for every  one has , where . John Tate developed this formula in his doctoral thesis to prove a more general version of the functional equation for the Riemann zeta function. This involves giving the zeta function of a number field an integral representation in which the integral of a Schwartz–Bruhat function, chosen as a test function, is twisted by a certain character and is integrated over  with respect to the multiplicative Haar measure of this group. This allows one to apply analytic methods to study zeta functions through these zeta integrals. References

Number theory
Topological groups