Peskin–Takeuchi parameter

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In mathematics, the Lehmer mean of a tuple x of positive real numbers, named after Derrick Henry Lehmer,[1] is defined as:

Lp(x)=k=1nxkpk=1nxkp1.

The weighted Lehmer mean with respect to a tuple w of positive weights is defined as:

Lp,w(x)=k=1nwkxkpk=1nwkxkp1.

The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.

Properties

The derivative of pLp(x) is non-negative

pLp(x)=j=1nk=j+1n(xjxk)(lnxjlnxk)(xjxk)p1(k=1nxkp1)2,

thus this function is monotonic and the inequality

pqLp(x)Lq(x)

holds.

Special cases

Sketch of a proof: Without loss of generality let x1,,xk be the values which equal the maximum. Then Lp(x)=x1k+(xk+1x1)p++(xnx1)pk+(xk+1x1)p1++(xnx1)p1

Applications

Signal processing

Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p. Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving Lehmer mean according to the following Haskell code.

 lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
 lehmerSmooth smooth p xs = zipWith (/)
                                     (smooth (map (**p) xs))
                                     (smooth (map (**(p-1)) xs))

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.