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

Examples

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

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

  • On the rational adeles A Q {\displaystyle \mathbb {A} _{\mathbb {Q} }} all functions in the Schwartz–Bruhat space S ( A Q ) {\displaystyle {\mathcal {S}}(\mathbb {A} _{\mathbb {Q} })} are finite linear combinations of ∏ p ≤ ∞ f p = f ∞ × ∏ p < ∞ f p {\displaystyle \prod _{p\leq \infty }f_{p}=f_{\infty }\times \prod _{p<\infty }f_{p}} over all rational primes p {\displaystyle p}, where f ∞ ∈ S ( R ) {\displaystyle f_{\infty }\in {\mathcal {S}}(\mathbb {R} )}, f p ∈ S ( Q p ) {\displaystyle f_{p}\in {\mathcal {S}}(\mathbb {Q} _{p})}, and f p = 1 Z p {\displaystyle f_{p}=\mathbf {1} _{\mathbb {Z} _{p}}} for all but finitely many p {\displaystyle p}. The sets Q p {\displaystyle \mathbb {Q} _{p}} and Z p {\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 A K {\displaystyle \mathbb {A} _{K}} the Schwartz–Bruhat space S ( A K ) {\displaystyle {\mathcal {S}}(\mathbb {A} _{K})} is dense in the space L 2 ( A K , d x ) . {\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 f ∈ S ( A K ) {\displaystyle f\in {\mathcal {S}}(\mathbb {A} _{K})} one has ∑ x ∈ K f ( a x ) = 1 | a | ∑ x ∈ K f ^ ( a − 1 x ) {\displaystyle \sum _{x\in K}f(ax)={\frac {1}{|a|}}\sum _{x\in K}{\hat {f}}(a^{-1}x)}, where a ∈ A K × {\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 A K × {\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.

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