Schwartz–Bruhat function

In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a 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 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, 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.
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.
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 we have
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 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.