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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,\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,\dots,x_n) = \left( \frac \sum_^n x_i^p \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, \dots, x_n) = \sqrt()
Furthermore, for a sequence of positive weights ''wi'' with sum \sum w_i = 1 we define the weighted power mean as:
:\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):
:\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,
:\lim_ \frac^p \right)}} = \lim_ \frac}} = \lim_ \frac} = \sum_^n w_i \ln = \ln \right)}
By the continuity of the exponential function, we can substitute back into the above relation to obtain
:\lim_ M_p(x_1,\dots,x_n) = \exp \right)} \right)} = \prod_^n x_i^ = M_0(x_1,\dots,x_n)
as desired.
|}
: M_p = M_
|-
|
Assume (possibly after relabeling and combining terms together) that x_1 \geq \dots \geq x_n. Then
:\lim_ M_p(x_1,\dots,x_n) = \lim_ \left( \sum_^n w_i x_i^p \right)^ = x_1 \lim_ \left( \sum_^n w_i \left( \frac \right)^p \right)^ = x_1 = M_\infty (x_1,\dots,x_n).
The formula for M_ follows from M_ (x_1,\dots,x_n) = \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



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