On a real vector space , the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space .
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 abelianlocally 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.[1]
On a general locally compact abelian group , let be a compactly generated subgroup, and a compact subgroup of such that is elementary. Then the pullback of a Schwartz–Bruhat function on is a Schwartz–Bruhat function on , and all Schwartz–Bruhat functions on are obtained like this for suitable and . (The space of Schwartz–Bruhat functions on is endowed with the inductive limit topology.)
In particular, on the ring of adeles over a global field, the Schwartz–Bruhat functions are finite linear combinations of the products over each place of , where each is a Schwartz–Bruhat function on a local field and is the characteristic function on the ring of integers for all but finitely many . (For the archimedean places of , the are just the usual Schwartz functions on , while for the non-archimedean places the are the Schwartz–Bruhat functions of non-archimedean local fields.)
The space of Schwartz–Bruhat functions on the adeles is defined to be the restricted tensor product[2] of Schwartz–Bruhat spaces of local fields, where is a finite set of places of . The elements of this space are of the form , where for all and for all but finitely many . For each we can write , which is finite and thus is well defined.[3]
Examples
Every Schwartz–Bruhat function can be written as , where each , , and .[4] This can be seen by observing that being a local field implies that by definition has compact support, i.e., has a finite subcover. Since every open set in can be expressed as a disjoint union of open balls of the form (for some and ) we have
. The function must also be locally constant, so for some . (As for evaluated at zero, is always included as a term.)
On the rational adeles all functions in the Schwartz–Bruhat space are finite linear combinations of over all rational primes , where , , and for all but finitely many . The sets and 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 the Schwartz–Bruhat space is dense in the space
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 one has , where . 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 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.[5]
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.
^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, ISBN978-0-9502734-2-6, MR 0217026
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.; et al. (1990). Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN0-12-279506-7.
Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge: Cambridge University Press. ISBN978-0521658188.
Ramakrishnan, V.; Valenza, R. J. (1999). Fourier Analysis on Number Fields. New York: Springer-Verlag. ISBN978-0387984360.
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, ISBN978-0-9502734-2-6, MR 0217026
January 07, 2023
schwartz, bruhat, function, mathematics, named, after, laurent, schwartz, françois, bruhat, complex, valued, function, locally, compact, abelian, group, such, adeles, that, generalizes, schwartz, function, real, vector, space, tempered, distribution, defined, . In mathematics a Schwartz Bruhat function named after Laurent Schwartz and Francois 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 Contents 1 Definitions 2 Examples 3 Properties 4 Applications 5 ReferencesDefinitions EditOn 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 1 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 2 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 3 Examples EditEvery 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 4 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 havesupp 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 lt f p displaystyle prod p leq infty f p f infty times prod p lt 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 EditThe 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 EditIn 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 5 References Edit 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 Bump p 300 Ramakrishnan Valenza p 260 Deitmar p 134 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 MR 0217026 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 et al 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 ISSN 0172 5939 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 MR 0217026 Retrieved from https en wikipedia org w index php title Schwartz Bruhat function amp oldid 1130480291, wikipedia, wiki, book, books, library,