Schwartz–Bruhat function

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.
• 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 $\bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)$ 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.

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 $c_{i}\in \mathbb {C}$. 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.

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.
• 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