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 $\mathbb^n$, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space $\mathcal(\mathbb^n)$.
*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 $\mathbb_K$ 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 $f_v = \mathbf_$ is the characteristic function on the ring of integers $\mathcal_v$ for all but finitely many $v$. (For the archimedean places of $K$, the $f_v$ are just the usual Schwartz functions on $\mathbb^n$, 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 $\mathbb_K$ is defined to be the restricted tensor product $\bigotimes_v'\mathcal(K_v) := \varinjlim_\left(\bigotimes_\mathcal(K_v) \right)$ of Schwartz–Bruhat spaces $\mathcal(K_v)$ 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 $f_v \in \mathcal(K_v)$ for all $v$ and $f_v, _=1$ for all but finitely many $v$. For each $x = (x_v)_v \in \mathbb_K$ we can write $f(x) = \prod_vf_v(x_v)$, which is finite and thus is well defined.

Examples

*Every Schwartz–Bruhat function $f \in \mathcal(\mathbb_p)$ can be written as $f = \sum_^n c_i \mathbf_$, where each $a_i \in \mathbb_p$, $k_i \in \mathbb$, and $c_i \in \mathbb$. This can be seen by observing that $\mathbb_p$ being a local field implies that $f$ by definition has compact support, i.e., $\operatorname(f)$ has a finite subcover. Since every open set in $\mathbb_p$ can be expressed as a disjoint union of open balls of the form $a + p^k \mathbb_p$ (for some $a \in \mathbb_p$ and $k \in \mathbb$) we have

: $\operatorname(f) = \coprod_^n (a_i + p^\mathbb_p)$. The function $f$ must also be locally constant, so $f , _ = c_i \mathbf_$ for some $c_i \in \mathbb$. (As for $f$ evaluated at zero, $f(0)\mathbf_$ is always included as a term.)

*On the rational adeles $\mathbb_$ all functions in the Schwartz–Bruhat space $\mathcal(\mathbb_)$ are finite linear combinations of $\prod_ f_p = f_\infty \times \prod_ f_p$ over all rational primes $p$, where $f_\infty \in \mathcal(\mathbb)$, $f_p \in \mathcal(\mathbb_p)$, and $f_p = \mathbf_$ for all but finitely many $p$. The sets $\mathbb_p$ and $\mathbb_p$ 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 $\mathbb_K$ the Schwartz–Bruhat space $\mathcal(\mathbb_K)$ is dense in the space $L^2(\mathbb_K, dx).$

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 $f \in \mathcal(\mathbb_K)$ one has $\sum_ f(ax) = \frac\sum_ \hat(a^x)$, where $a \in \mathbb_K^$. 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 $\mathbb_K^$ 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

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{{DEFAULTSORT:Schwartz-Bruhat function
Number theory
Topological groups