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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hölder mean ： ウィキペディア英語版 Generalized mean In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). The generalized mean is also known as power mean or Hölder mean (named after Otto Hölder). ==Definition== If ''p'' is a non-zero real number, and $x\_1,\backslash dots,x\_n$ are positive real numbers, then the generalized mean or power mean with exponent ''p'' of these positive real numbers is: :$M\_p(x\_1,\backslash dots,x\_n)\; =\; \backslash left(\; \backslash frac\; \backslash sum\_^n\; x\_i^p\; \backslash right)^\}\; .$ Note the relationship to the p-norm. For ''p'' = 0 we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case): :$M\_0(x\_1,\; \backslash dots,\; x\_n)\; =\; \backslash sqrt()$ Furthermore, for a sequence of positive weights ''wi'' with sum $\backslash sum\; w\_i\; =\; 1$ we define the weighted power mean as: :$\backslash begin$ M_p(x_1,\dots,x_n) &= \left(\sum_^n w_i x_i^p \right)^} \\ M_0(x_1,\dots,x_n) &= \prod_^n x_i^ \end The unweighted means correspond to setting all ''wi'' = 1/''n''. For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below): :$\backslash begin$ M_(x_1, \dots, x_n) &= \max(x_1, \dots, x_n) \\ M_(x_1, \dots, x_n) &= \min(x_1, \dots, x_n) \end :^n" TITLE="\left(\sum_^n">w_ix_^p \right)^\right )} \right) } = \exp^p \right)}} \right) } In the limit ''p'' → 0, we can apply L'Hôpital's rule, differentiating the numerator and denominator with respect to p, to the argument of the exponential function, :$\backslash lim\_\; \backslash frac^p\; \backslash right)\}\}\; =\; \backslash lim\_\; \backslash frac\}\}\; =\; \backslash lim\_\; \backslash frac\}\; =\; \backslash sum\_^n\; w\_i\; \backslash ln\; =\; \backslash ln\; \backslash right)\}$ By the continuity of the exponential function, we can substitute back into the above relation to obtain :$\backslash lim\_\; M\_p(x\_1,\backslash dots,x\_n)\; =\; \backslash exp\; \backslash right)\}\; \backslash right)\}\; =\; \backslash prod\_^n\; x\_i^\; =\; M\_0(x\_1,\backslash dots,x\_n)$ as desired. |} : M_p = M_ |- | Assume (possibly after relabeling and combining terms together) that $x\_1\; \backslash geq\; \backslash dots\; \backslash geq\; x\_n$. Then :$\backslash lim\_\; M\_p(x\_1,\backslash dots,x\_n)\; =\; \backslash lim\_\; \backslash left(\; \backslash sum\_^n\; w\_i\; x\_i^p\; \backslash right)^\; =\; x\_1\; \backslash lim\_\; \backslash left(\; \backslash sum\_^n\; w\_i\; \backslash left(\; \backslash frac\; \backslash right)^p\; \backslash right)^\; =\; x\_1\; =\; M\_\backslash infty\; (x\_1,\backslash dots,x\_n).$ The formula for $M\_$ follows from $M\_\; (x\_1,\backslash dots,x\_n)\; =\; \backslash frac.$ |} 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「Generalized mean」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.