Schwartz–Bruhat function explained

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

R ^{n} , the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space l{S}( R ^{n)} .

• 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.^[1]
• 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 endowed with the inductive limit topology.)K , a Schwartz–Bruhat function is a locally constant function of compact support.

• In particular, on the ring of adeles

A _{K} over a global field K , the Schwartz–Bruhat functions f are finite linear combinations of the products \prod _{v} f _{v} over each place v of K , where each f _{v} is a Schwartz–Bruhat function on a local field K _{v} and f _{v} = 1 _{l{O} _{v}} is the characteristic function on the ring of integers l{O} _{v} for all but finitely many v . (For the archimedean places of K , the f _{v} are just the usual Schwartz functions on R ^{n} , while for the non-archimedean places the f _{v} are the Schwartz–Bruhat functions of non-archimedean local fields.)

• The space of Schwartz–Bruhat functions on the adeles

A _{K} is defined to be the restricted tensor product^[2] o times _{v'l{S}(K} _{v)} : = \varinjlim _{E} \left( o times _{v} l{S}(K _{v)} \right) of Schwartz–Bruhat spaces l{S}(K _{v)} of local fields, where E is a finite set of places of K . The elements of this space are of the form f = ⊗ _{vf} _{v} , where f _{v} \in l{S}(K _{v)} for all v and f _{v|} _{l{O} _{v}=1} for all but finitely many v . For each x = (x _{v)} _{v} \in A _{K} we can write f(x) = \prod _{vf} _{v(x} _{v)} , which is finite and thus is well defined.^[3]

Examples

• Every Schwartz–Bruhat function

f \in l{S}( Q _{p)} can be written as f = \sum ^{n} _{i =1} c _{i} 1 _{a _{i} + p ^{k _{i}} Z _{p}} , where each a _{i} \in Q _{p} , k _{i} \in Z , and c _{i} \in C .^[4] This can be seen by observing that Q _{p} being a local field implies that f by definition has compact support, i.e., \operatorname{supp}(f) has a finite subcover. Since every open set in Q _{p} can be expressed as a disjoint union of open balls of the form a + p ^{k} Z _{p} (for some a \in Q _{p} and k \in Z ) we have\operatorname{supp}(f) = \coprod ^{n} _{i =1} (a _{i} + p ^{k _{i}} Z _{p)} . The function f must also be locally constant, so f | _{a _{i} + p ^{k _{i}} Z _{p}} = c _{i} 1 _{a _{i} + p ^{k _{i}} Z _{p}} for some c _{i} \in C . (As for f evaluated at zero, f( 0 ) 1 _{Z _{p}} is always included as a term.)

• On the rational adeles

A _{Q} all functions in the Schwartz–Bruhat space l{S}( A _{Q} ) are finite linear combinations of \prod _{p} f _{p} = f _{infty} x \prod _{p} f _{p} over all rational primes p , where f _{infty} \in l{S}( R ) , f _{p} \in l{S}( Q _{p)} , and f _{p} = 1 _{Z _{p}} for all but finitely many p . The sets Q _{p} and Z _{p} 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

A _{K} the Schwartz–Bruhat space l{S}( A _{K)} is dense in the space L ^{2 ( A} _{K,} dx).

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

f \in l{S}( A _{K)} one has \sum _{x} f(ax) = (1)/(|a|) \sum _{x} \hat{f}(a ^{-1} x) , where a \in A ^{x} _{K} . 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 A ^{x} _{K} 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

• Osborne . M. Scott . On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups . Journal of Functional Analysis . 19 . 1975 . 40–49 . 10.1016/0022-1236(75)90005-1 . free .
• Book: Gelfand, I. M. . Representation Theory and Automorphic Functions . Boston . Academic Press . 1990 . 0-12-279506-7 . etal.
• Book: Bump, Daniel . Automorphic Forms and Representations. Cambridge. Cambridge University Press. 1998. 978-0521658188.
• Book: Deitmar, Anton . Automorphic Forms. Berlin. Springer-Verlag London . 2012 . 978-1-4471-4434-2. 0172-5939.
• Book: Ramakrishnan . V. . Valenza . R. J.. Fourier Analysis on Number Fields. New York. Springer-Verlag. 1999. 978-0387984360.

Notes and References

1. Osborne . M. Scott . On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups . Journal of Functional Analysis . 19 . 1975 . 40–49 . 10.1016/0022-1236(75)90005-1 . free .
2. Bump, p.300
3. Ramakrishnan, Valenza, p.260
4. Deitmar, p.134