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.
Contents
Definitions
On a real vector space
R
n
{\displaystyle \mathbb {R} ^{n}}
, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space
S
(
R
n
)
{\displaystyle {\mathcal {S}}(\mathbb {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.
On a general locally compact abelian group
G
{\displaystyle G}
, let
A
{\displaystyle A}
be a compactly generated subgroup, and
B
{\displaystyle B}
a compact subgroup of
A
{\displaystyle A}
such that
A
/
B
{\displaystyle A/B}
is elementary. Then the pullback of a Schwartz–Bruhat function on
A
/
B
{\displaystyle A/B}
is a Schwartz–Bruhat function on
G
{\displaystyle G}
, and all Schwartz–Bruhat functions on
G
{\displaystyle G}
are obtained like this for suitable
A
{\displaystyle A}
and
B
{\displaystyle B}
. (The space of Schwartz–Bruhat functions on
G
{\displaystyle G}
is endowed with the inductive limit topology.)
On a non-archimedean local field
K
{\displaystyle K}
, a Schwartz–Bruhat function is a locally constant function of compact support.
In particular, on the ring of adeles
A
K
{\displaystyle \mathbb {A} _{K}}
over a global field
K
{\displaystyle K}
, the Schwartz–Bruhat functions
f
{\displaystyle f}
are finite linear combinations of the products
∏
v
f
v
{\displaystyle \prod _{v}f_{v}}
over each place
v
{\displaystyle v}
of
K
{\displaystyle K}
, where each
f
v
{\displaystyle f_{v}}
is a Schwartz–Bruhat function on a local field
K
v
{\displaystyle K_{v}}
and
f
v
=
1
O
v
{\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}
is the characteristic function on the ring of integers
O
v
{\displaystyle {\mathcal {O}}_{v}}
for all but finitely many
v
{\displaystyle v}
. (For the archimedean places of
K
{\displaystyle K}
, the
f
v
{\displaystyle f_{v}}
are just the usual Schwartz functions on
R
n
{\displaystyle \mathbb {R} ^{n}}
, while for the non-archimedean places the
f
v
{\displaystyle f_{v}}
are the Schwartz–Bruhat functions of non-archimedean local fields.)
The space of Schwartz–Bruhat functions on the adeles
A
K
{\displaystyle \mathbb {A} _{K}}
is defined to be the restricted tensor product
⨂
v
′
S
(
K
v
)
:=
lim
→
E
(
⨂
v
∈
E
S
(
K
v
)
)
{\displaystyle \bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)}
of Schwartz–Bruhat spaces
S
(
K
v
)
{\displaystyle {\mathcal {S}}(K_{v})}
of local fields, where
E
{\displaystyle E}
is a finite set of places of
K
{\displaystyle K}
. The elements of this space are of the form
f
=
⊗
v
f
v
{\displaystyle f=\otimes _{v}f_{v}}
, where
f
v
∈
S
(
K
v
)
{\displaystyle f_{v}\in {\mathcal {S}}(K_{v})}
for all
v
{\displaystyle v}
and
f
v
=
1
O
v
{\displaystyle f_{v}=1_{{\mathcal {O}}_{v}}}
for all but finitely many
v
{\displaystyle v}
. For each
x
=
(
x
v
)
v
∈
A
K
{\displaystyle x=(x_{v})_{v}\in \mathbb {A} _{K}}
we can write
f
(
x
)
=
∏
v
f
v
(
x
v
)
{\displaystyle f(x)=\prod _{v}f_{v}(x_{v})}
, which is finite and thus is well defined.
Examples
Every Schwartz–Bruhat function
f
∈
S
(
Q
p
)
{\displaystyle f\in {\mathcal {S}}(\mathbb {Q} _{p})}
can be written as
f
=
∑
i
=
1
n
c
i
1
a
i
+
p
k
i
Z
p
{\displaystyle f=\sum _{i=1}^{n}c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}}
, where each
a
i
∈
Q
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
{\displaystyle \mathbb {A} _{K}}
the Schwartz–Bruhat space
S
(
A
K
)
{\displaystyle {\mathcal {S}}(\mathbb {A} _{K})}
is dense in the space
L
2
(
A
K
,
d
x
)
.
{\displaystyle L^{2}(\mathbb {A} _{K},dx).}
Fourier transform
Let
G
{\displaystyle G}
be a locally compact abelian group and let
G
^
{\displaystyle {\widehat {G}}}
denote its Pontryagin dual. After choosing compatible Haar measures on
G
{\displaystyle G}
and
G
^
{\displaystyle {\widehat {G}}}
, the Fourier transform of an integrable function
f
{\displaystyle f}
on
G
{\displaystyle G}
is defined by
f
^
(
χ
)
=
∫
G
f
(
Topology
The Schwartz–Bruhat space is not only a vector space of functions but also a locally convex topological vector space. In the real case this is the usual Fréchet topology on the Schwartz space, defined by the Schwartz seminorms. In the non-archimedean local-field case, the space is an inductive limit of finite-dimensional spaces of functions supported on a fixed compact open set and invariant under translation by a fixed compact open subgroup. The continuity condition in the definition of a tempered distribution refers to this locally convex topology.
With this locally convex topology, Schwartz–Bruhat spaces are nuclear spaces. In particular, the usual examples include the nuclear Fréchet spaces
S
(
R
n
)
{\displaystyle S(\mathbf {R} ^{n})}
,
C
∞
(
T
n
)
{\displaystyle C^{\infty }(\mathbf {T} ^{n})}
, and the space of rapidly decreasing functions on
Z
n
{\displaystyle \mathbf {Z} ^{n}}
; in the non-archimedean local-field case,
S
(
Tempered distributions
The continuous dual of a Schwartz–Bruhat space is the corresponding space of tempered distributions. Thus a tempered distribution on a locally compact abelian group
G
{\displaystyle G}
is a continuous linear functional on
S
(
G
)
{\displaystyle S(G)}
. Since the Fourier transform is an isomorphism from
S
(
G
)
{\displaystyle S(G)}
to
S
(
G
^
)
{\displaystyle S({\widehat {G}})}
, it extends by duality to an isomorphism between tempered distributions on
G
{\displaystyle G}
and tempered distributions on
G
^
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
∈
S
(
A
K
)
{\displaystyle f\in {\mathcal {S}}(\mathbb {A} _{K})}
one has
∑
x
∈
K
f
(
a
x
)
=
1
|
a
|
∑
x
∈
K
f
^
(
a
−
1