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.

• 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.^[1]
• 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^[2] ${\displaystyle \bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)}$ 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.^[3]

• 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 ${\displaystyle c_{i}\in \mathbb {C} }$.^[4] 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.

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 ${\displaystyle L^{2}(\mathbb {A} _{K},dx).}$

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.^[5]

• 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