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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 $\mathbb {R} ^{n}$ , the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space ${\mathcal {S}}(\mathbb {R} ^{n})$ .
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.^[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.)
On a non-archimedean local field $K$ , a Schwartz–Bruhat function is a locally constant function of compact support.
In particular, on the ring of adeles $\mathbb {A} _{K}$ 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 $f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}$ is the characteristic function on the ring of integers ${\mathcal {O}}_{v}$ for all but finitely many $v$ . (For the archimedean places of $K$ , the $f_{v}$ are just the usual Schwartz functions on $\mathbb {R} ^{n}$ , 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 $\mathbb {A} _{K}$ is defined to be the restricted tensor product^[2] $\bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)$ of Schwartz–Bruhat spaces ${\mathcal {S}}(K_{v})$ of local fields, where $E$ is a finite set of places of $K$ . The elements of this space are of the form $f=\otimes _{v}f_{v}$ , where $f_{v}\in {\mathcal {S}}(K_{v})$ for all $v$ and $f_{v}|_{{\mathcal {O}}_{v}}=1$ for all but finitely many $v$ . For each $x=(x_{v})_{v}\in \mathbb {A} _{K}$ we can write $f(x)=\prod _{v}f_{v}(x_{v})$ , which is finite and thus is well defined.^[3]

Examples

Every Schwartz–Bruhat function $f\in {\mathcal {S}}(\mathbb {Q} _{p})$ can be written as $f=\sum _{i=1}^{n}c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}$ , where each $a_{i}\in \mathbb {Q} _{p}$ , $k_{i}\in \mathbb {Z}$ , and $c_{i}\in \mathbb {C}$ .^[4] This can be seen by observing that $\mathbb {Q} _{p}$ 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 $\mathbb {Q} _{p}$ can be expressed as a disjoint union of open balls of the form $a+p^{k}\mathbb {Z} _{p}$ (for some $a\in \mathbb {Q} _{p}$ and $k\in \mathbb {Z}$ ) we have

\operatorname {supp} (f)=\coprod _{i=1}^{n}(a_{i}+p^{k_{i}}\mathbb {Z} _{p})

. The function

f

must also be locally constant, so

f|_{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}=c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}

for some

c_{i}\in \mathbb {C}

. (As for

f

evaluated at zero,

f(0)\mathbf {1} _{\mathbb {Z} _{p}}

is always included as a term.)

On the rational adeles $\mathbb {A} _{\mathbb {Q} }$ all functions in the Schwartz–Bruhat space ${\mathcal {S}}(\mathbb {A} _{\mathbb {Q} })$ are finite linear combinations of $\prod _{p\leq \infty }f_{p}=f_{\infty }\times \prod _{p<\infty }f_{p}$ over all rational primes $p$ , where $f_{\infty }\in {\mathcal {S}}(\mathbb {R} )$ , $f_{p}\in {\mathcal {S}}(\mathbb {Q} _{p})$ , and $f_{p}=\mathbf {1} _{\mathbb {Z} _{p}}$ for all but finitely many $p$ . The sets $\mathbb {Q} _{p}$ and $\mathbb {Z} _{p}$ 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 $\mathbb {A} _{K}$ the Schwartz–Bruhat space ${\mathcal {S}}(\mathbb {A} _{K})$ is dense in the space $L^{2}(\mathbb {A} _{K},dx).$

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 $f\in {\mathcal {S}}(\mathbb {A} _{K})$ one has $\sum _{x\in K}f(ax)={\frac {1}{|a|}}\sum _{x\in K}{\hat {f}}(a^{-1}x)$ , where $a\in \mathbb {A} _{K}^{\times }$ . 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 $\mathbb {A} _{K}^{\times }$ 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.^[5]

References

^ 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.
^ Bump, p.300
^ Ramakrishnan, Valenza, p.260
^ Deitmar, p.134
^ 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

[1] 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.

[2] Bump, p.300

[3] Ramakrishnan, Valenza, p.260

[4] Deitmar, p.134

[5] 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

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