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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. ==Definitions== *On a real vector space, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing). *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 topologized with the inductive limit topology.) *In particular, on the ring of adeles over a number field or function field, the Schwartz–Bruhat functions are linear combinations of products of Schwartz functions on the infinite part and locally constant functions of compact support at the non-archimedean places (equal to the characteristic function of the integers at all but a finite number of places). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schwartz–Bruhat function」の詳細全文を読む スポンサード リンク
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