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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 ${\displaystyle \mathbb {R} ^{n}}$, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$.
• 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 ${\displaystyle G}$, let ${\displaystyle A}$ be a compactly generated subgroup, and ${\displaystyle B}$ a compact subgroup of ${\displaystyle A}$ such that ${\displaystyle A/B}$ is elementary. Then the pullback of a Schwartz–Bruhat function on ${\displaystyle A/B}$ is a Schwartz–Bruhat function on ${\displaystyle G}$, and all Schwartz–Bruhat functions on ${\displaystyle G}$ are obtained like this for suitable ${\displaystyle A}$ and ${\displaystyle B}$. (The space of Schwartz–Bruhat functions on ${\displaystyle G}$ is endowed with the inductive limit topology.)
• On a non-archimedean local field ${\displaystyle K}$, a Schwartz–Bruhat function is a locally constant function of compact support.
• In particular, on the ring of adeles ${\displaystyle \mathbb {A} _{K}}$ over a global field ${\displaystyle K}$, the Schwartz–Bruhat functions ${\displaystyle f}$ are finite linear combinations of the products ${\displaystyle \prod _{v}f_{v}}$ over each place ${\displaystyle v}$ of ${\displaystyle K}$, where each ${\displaystyle f_{v}}$ is a Schwartz–Bruhat function on a local field ${\displaystyle K_{v}}$ and ${\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}$ is the characteristic function on the ring of integers ${\displaystyle {\mathcal {O}}_{v}}$ for all but finitely many ${\displaystyle v}$. (For the archimedean places of ${\displaystyle K}$, the ${\displaystyle f_{v}}$ are just the usual Schwartz functions on ${\displaystyle \mathbb {R} ^{n}}$, while for the non-archimedean places the ${\displaystyle f_{v}}$ are the Schwartz–Bruhat functions of non-archimedean local fields.)
• The space of Schwartz–Bruhat functions on the adeles ${\displaystyle \mathbb {A} _{K}}$ is defined to be the restricted tensor product ${\displaystyle \bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)}$ of Schwartz–Bruhat spaces ${\displaystyle {\mathcal {S}}(K_{v})}$ of local fields, where ${\displaystyle E}$ is a finite set of places of ${\displaystyle K}$. The elements of this space are of the form ${\displaystyle f=\otimes _{v}f_{v}}$, where ${\displaystyle f_{v}\in {\mathcal {S}}(K_{v})}$ for all ${\displaystyle v}$ and ${\displaystyle f_{v}|_{{\mathcal {O}}_{v}}=1}$ for all but finitely many ${\displaystyle v}$. For each ${\displaystyle x=(x_{v})_{v}\in \mathbb {A} _{K}}$ we can write ${\displaystyle f(x)=\prod _{v}f_{v}(x_{v})}$, which is finite and thus is well defined.

Examples

• Every Schwartz–Bruhat function ${\displaystyle f\in {\mathcal {S}}(\mathbb {Q} _{p})}$ can be written as ${\displaystyle f=\sum _{i=1}^{n}c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}}$, where each ${\displaystyle a_{i}\in \mathbb {Q} _{p}}$, ${\displaystyle k_{i}\in \mathbb {Z} }$, and ${\displaystyle c_{i}\in \mathbb {C} }$. This can be seen by observing that ${\displaystyle \mathbb {Q} _{p}}$ being a local field implies that ${\displaystyle f}$ by definition has compact support, i.e., ${\displaystyle \operatorname {supp} (f)}$ has a finite subcover. Since every open set in ${\displaystyle \mathbb {Q} _{p}}$ can be expressed as a disjoint union of open balls of the form ${\displaystyle a+p^{k}\mathbb {Z} _{p}}$ (for some ${\displaystyle a\in \mathbb {Q} _{p}}$ and ${\displaystyle k\in \mathbb {Z} }$) we have

${\displaystyle \operatorname {supp} (f)=\coprod _{i=1}^{n}(a_{i}+p^{k_{i}}\mathbb {Z} _{p})}$. The function ${\displaystyle f}$ must also be locally constant, so ${\displaystyle f|_{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}=c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}}$ for some ${\displaystyle c_{i}\in \mathbb {C} }$. (As for ${\displaystyle f}$ evaluated at zero, ${\displaystyle f(0)\mathbf {1} _{\mathbb {Z} _{p}}}$ is always included as a term.)

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

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

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

• 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

• Number theory
• Topological groups