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.
- 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.[1]
- 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[2]
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.[3]
- Every Schwartz–Bruhat function
can be written as
, where each
,
, and
.[4] 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.
Ramakrishnan, Valenza, p.260
Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026
- Osborne, M. Scott (1975). "On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups". Journal of 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.
- Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge: Cambridge University Press. ISBN 978-0521658188.
- Deitmar, Anton (2012). Automorphic Forms. Berlin: Springer-Verlag London. ISBN 978-1-4471-4434-2. ISSN 0172-5939.
- Ramakrishnan, V.; Valenza, R. J. (1999). Fourier Analysis on Number Fields. New York: Springer-Verlag. ISBN 978-0387984360.
- Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026