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In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.〔See .〕 They provide a necessary and sufficient condition under which any function in can be approximated by linear combinations of translations of a given function.〔see .〕 Informally, if the Fourier transform of a function vanishes on a certain set , the Fourier transform of any linear combination of translations of also vanishes on . Therefore, the linear combinations of translations of can not approximate a function whose Fourier transform does not vanish on . Wiener's theorems make this precise, stating that linear combinations of translations of are dense if and only the zero set of the Fourier transform of is empty (in the case of ) or of Lebesgue measure zero (in the case of ). Gelfand reformulated Wiener's theorem in terms of commutative C *-algebras, when it states that the spectrum of the L1 group ring L1(R) of the group R of real numbers is the dual group of R. A similar result is true when R is replaced by any locally compact abelian group. ==The condition in == Let be an integrable function. The span of translations = is dense in if and only if the Fourier transform of has no real zeros. 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「Wiener's tauberian theorem」の詳細全文を読む スポンサード リンク
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