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In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by . It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group. ==Definition== The definition of the Schwartz space uses Harish-Chandra's Ξ function and his ''σ'' function. The ''σ'' function is defined by : for ''x''=''k'' exp ''X'' with ''k'' in ''K'' and ''X'' in ''p'' for a Cartan decomposition ''G'' = ''K'' exp ''p'' of the Lie group ''G'', where ||''X''|| is a ''K''-invariant Euclidean norm on ''p'', usually chosen to be the Killing form. . The Schwartz space on ''G'' consists roughly of the functions all of whose derivatives are rapidly decreasing compared to ''Ξ''. More precisely, if ''G'' is connected then the Schwartz space consists of all smooth functions ''f'' on ''G'' such that : is bounded, where ''D'' is a product of left-invariant and right-invariant differential operators on ''G'' . 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「Harish-Chandra's Schwartz space」の詳細全文を読む スポンサード リンク
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