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.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.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 everyJohn 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 overwith respect to the multiplicative Haar measure of this group.
mathematicsLaurent SchwartzFrançois Bruhatlocally compact abelian groupadelesSchwartz functionabelianlocally compact groupintegerscircle groupcompactly generatedinductive limit topologylocal fieldlocally constant functionglobal fieldcharacteristic functionring of integersp-adic numbersp-adic integersFourier transformPontryagin dualalgebraic number theoryPoisson summation formulaJohn Tatedoctoral thesisRiemann zeta function