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 ${𝒮}({\mathbb{ℝ}}^n)$.
• 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 $f_v={\mathbf{1}}_{{{𝒪}}_v}$ is the characteristic function on the ring of integers ${{𝒪}}_v$ 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 ${\underset{v}{⨂}}^{\prime }{𝒮}(K_v):=\underset{E}{\underrightarrow{\lim }}(\underset{v\in E}{⨂}{𝒮}(K_v))$ of Schwartz–Bruhat spaces ${𝒮}(K_v)$ 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 $f_v\in {𝒮}(K_v)$ for all $v$ and $f_v{|}_{{{𝒪}}_v}=1$ 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 $f\in {𝒮}({\mathbb{\mathbb{Q}}}_p)$ can be written as $f=\sum _{i=1}^n{c}_i{\mathbf{1}}_{a_i+p^{k_i}{\mathbb{\mathbb{Z}}}_p}$, where each $a_i\in {\mathbb{\mathbb{Q}}}_p$, $k_i\in \mathbb{\mathbb{Z}}$, and $c_i\in \mathbb{ℂ}$. This can be seen by observing that ${\mathbb{\mathbb{Q}}}_p$ being a local field implies that $f$ by definition has compact support, i.e., $\mathrm{supp}(f)$ has a finite subcover. Since every open set in ${\mathbb{\mathbb{Q}}}_p$ can be expressed as a disjoint union of open balls of the form $a+p^k{\mathbb{\mathbb{Z}}}_p$ (for some $a\in {\mathbb{\mathbb{Q}}}_p$ and $k\in \mathbb{\mathbb{Z}}$) we have

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

• On the rational adeles ${\mathbb{𝔸}}_{\mathbb{\mathbb{Q}}}$ all functions in the Schwartz–Bruhat space ${𝒮}({\mathbb{𝔸}}_{\mathbb{\mathbb{Q}}})$ are finite linear combinations of $\prod _{p\le \mathrm{\infty }}{f}_p=f_{\mathrm{\infty }}\times \prod _{p<\mathrm{\infty }}{f}_p$ over all rational primes $p$, where $f_{\mathrm{\infty }}\in {𝒮}(\mathbb{ℝ})$, $f_p\in {𝒮}({\mathbb{\mathbb{Q}}}_p)$, and $f_p={\mathbf{1}}_{{\mathbb{\mathbb{Z}}}_p}$ for all but finitely many $p$. The sets ${\mathbb{\mathbb{Q}}}_p$ and ${\mathbb{\mathbb{Z}}}_p$ 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 ${𝒮}({\mathbb{𝔸}}_K)$ 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 $f\in {𝒮}({\mathbb{𝔸}}_K)$ 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