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.{{Cite journal |last=Osborne |first=M. Scott |title=On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups |journal=Journal of Functional Analysis |volume=19 |year=1975 |pages=40–49 |doi=10.1016/0022-1236(75)90005-1 |doi-access=free }}
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 productBump, p.300 $\bigotimes_v'\mathcal{S}(K_v) := \varinjlim_{E}\left(\bigotimes_{v \in E}\mathcal{S}(K_v) \right)$ 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 $f = \otimes_vf_v$ , 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 $f(x) = \prod_vf_v(x_v)$ , which is finite and thus is well defined.Ramakrishnan, Valenza, p.260

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 $c_i \in \mathbb{C}$ .Deitmar, p.134 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 $\prod_{p \le \infty} f_p = f_\infty \times \prod_{p < \infty } f_p$ 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 $\sum_{x \in K} f(ax) = \frac{1}$

\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.{{Citation | last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305–347}}

References

{{Cite journal |last=Osborne |first=M. Scott |title=On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups |journal=Journal of Functional Analysis |volume=19 |year=1975 |pages=40–49 |doi=10.1016/0022-1236(75)90005-1 |doi-access=free }}
{{Cite book |last=Gelfand |first=I. M. |title=Representation Theory and Automorphic Functions |location=Boston |publisher=Academic Press |year=1990 |isbn=0-12-279506-7 |display-authors=etal}}
{{Cite book |last = Bump |first = Daniel |title=Automorphic Forms and Representations| location=Cambridge| publisher=Cambridge University Press| year=1998| isbn=978-0521658188}}
{{Cite book |last=Deitmar |first=Anton |title=Automorphic Forms| location=Berlin| publisher=Springer-Verlag London |year=2012 |isbn=978-1-4471-4434-2| issn=0172-5939}}
{{Cite book | last1=Ramakrishnan | first1=V. | last2=Valenza | first2=R. J.| title=Fourier Analysis on Number Fields| location=New York| publisher=Springer-Verlag| year=1999| isbn=978-0387984360}}
{{Citation | last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305–347}}

Category:Number theory

Category:Topological groups

Category:Bruhat family