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High Quality Content by WIKIPEDIA articles! In mathematics, a Schwartz Bruhat function, named after Laurent Schwartz and François Bruhat, is a 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. 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.