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.
 * 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 product $$\bigotimes_v'\mathcal{S}(K_v) := \varinjlim_{E}\left(\bigotimes_{v \in E}\mathcal{S}(K_v) \right) $$ 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.

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 $$ c_i \in \mathbb{C} $$. 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 $$ \sum_{x \in K} f(ax) = \frac{1}{|a|}\sum_{x \in K} \hat{f}(a^{-1}x) $$, 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.