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In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem states that every multivariable continuous function can be represented as a superposition of continuous functions of two variables. It solved a version of Hilbert's thirteenth problem.〔Shigeo Akashi (2001). "Application of ϵ-entropy theory to Kolmogorov—Arnold representation theorem", ''Reports on Mathematical Physics'', v. 48, pp. 19–26 doi:10.1016/S0034-4877(01)80060-4〕 The works of Kolmogorov and Arnold established that if ''f'' is a multivariate continuous function, then ''f'' can be written as a finite composition of continuous functions of a single variable and the binary operation of addition.〔Dror Bar-Natan, ''Dessert: Hilbert's 13th Problem, in Full Colour'' ((link ))〕 For example:〔 : ''f''(''x'',''y'') = ''xy'' can be written as ''f''(''x'',''y'') = exp(log ''x'' + log ''y'') : ''f''(''x'',''y'',''z'') = ''x''''y'' / ''z'' can be written as ''f''(''x'', ''y'', ''z'') = exp(exp(log ''y'' + log log ''x'') + (−log ''z'')) In a sense, they showed that the only true multivariate function is the sum, since every other function can be written using univariate functions and summing.〔Persi Diaconis and Mehrdad Shahshahani, ''On Linear Functions of Linear Combinations'' (1984) p. 180 ((link ))〕 ==Original references== *A. N. Kolmogorov, "On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables", ''Dokl. Akad. Nauk SSSR'', 108 (1956), pp. 179–182; English translation: ''Amer. Math. Soc. Transl.'', 17 (1961), pp. 369–373. *V. I. Arnold, "On functions of three variables", ''Dokl. Akad. Nauk SSSR'', 114 (1957), pp. 679–681; English translation: ''Amer. Math. Soc. Transl.'', 28 (1963), pp. 51–54. 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「Kolmogorov–Arnold representation theorem」の詳細全文を読む スポンサード リンク
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