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

Examples

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

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

• On the rational adeles [math]\displaystyle{ \mathbb{A}_{\mathbb{Q}} }[/math] all functions in the Schwartz–Bruhat space [math]\displaystyle{ \mathcal{S}(\mathbb{A}_{\mathbb{Q}}) }[/math] are finite linear combinations of [math]\displaystyle{ \prod_{p \le \infty} f_p = f_\infty \times \prod_{p \lt \infty } f_p }[/math] over all rational primes [math]\displaystyle{ p }[/math], where [math]\displaystyle{ f_\infty \in \mathcal{S}(\mathbb{R}) }[/math], [math]\displaystyle{ f_p \in \mathcal{S}(\mathbb{Q}_p) }[/math], and [math]\displaystyle{ f_p = \mathbf{1}_{\mathbb{Z}_p} }[/math] for all but finitely many [math]\displaystyle{ p }[/math]. The sets [math]\displaystyle{ \mathbb{Q}_p }[/math] and [math]\displaystyle{ \mathbb{Z}_p }[/math] 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 [math]\displaystyle{ \mathbb{A}_K }[/math] the Schwartz–Bruhat space [math]\displaystyle{ \mathcal{S}(\mathbb{A}_K) }[/math] is dense in the space [math]\displaystyle{ L^2(\mathbb{A}_K, dx). }[/math]

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 [math]\displaystyle{ f \in \mathcal{S}(\mathbb{A}_K) }[/math] one has [math]\displaystyle{ \sum_{x \in K} f(ax) = \frac{1}{|a|}\sum_{x \in K} \hat{f}(a^{-1}x) }[/math], where [math]\displaystyle{ a \in \mathbb{A}_K^{\times} }[/math]. 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 [math]\displaystyle{ \mathbb{A}_K^{\times} }[/math] 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. (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. 
• 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