Robert J. MacG. Dawson writes:
> computing the arithmetic mean, and then exponentiating. (For
> mathematicians, it's the conjugate of the arithmetic mean under the
> log transform; note that the exponential transformation "undoes" the
> log transform.)
Thanks for the great article. I got interested in this concept,
conjugate of something under a transformation, but failed to find out
more about it. Where should I look?
Google gave a lot of hits for `conjugate under transform', but none of
complex conjugate, under GPL, Fourier transform, ... seemed relevant.
What I worked out is, various "means" can be taken to be functions
from something like vectors (of varying lengths) to real numbers as
follows. This is indeed elegant.
A (x1,...,xn) = sum_k (1/n) xk (arithmetic mean)
G = exp o A o log (geometric mean)
H = reciprocal o A o reciprocal (harmonic mean)
R = squareroot o A o square (root-mean-square)
And in general, I would now take `conjugate of X under transform f' to
mean f^-1 o X o f where f is applied to elements of vector, X takes a
vector to a scalar, and f^-1 "undoes" f.
Is this all right? Is there a better way to understand this concept?
What else might I find in place of X and f?
Thanks for any wisdom,
--
Jussi
.
.
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