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   -> Volume 4, Issue 4

Question: self-similar sum of arbitrary function
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Amy Caplan

PostPosted: Mon Dec 02, 2002 6:07 pm    
Subject: Question: self-similar sum of arbitrary function
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Question: self-similar sum of arbitrary function

I am curious about under what conditions the sum of a translated,
scaled "basis" function g() will tend to generate a statistically
self-similar function, as k->inf:
f(t) = sum_k w[k]*g(s[k]*t-o[k]),
where o[k] translates and s[k] and w[k] stretch and scale the function g().

Does anyone have any guidance to offer about how to reason about this?

Suppose s[k] are all 1. It seems obvious to me (thinking 'spectrally')
that no possible w[k] will generate a self-similar image
(unless g() already has a 1/f type spectrum. let's assume it does not).

Allow s[k] to change: increasing s[k] will shrink g() and create a
proportional bandwidth increase; the peak frequency amplitude will
also go down so as to maintain constant "energy".
If you plot the spectrum of different g() with s() taking on many values
it looks like the envelope of these spectra are already 1/f.
But, what is the distribution of s[k] needed to make this true?

Intuitively, it seems to me that
- the positioning o[k] does not affect the spectrum and can be random.
- if w[k] = 1 and s[k] are chosen to be uniformly distributed values,
then the spectrum might be 1/f. The function will be 'wierd' however--
isolated copies of highly-shrunken g() will poke up in random places.
- if s[k] are chosen proportionally, e.g. twice as many s[k]=2 as s[k]=1
and the corresponding w[k] falloff correspondingly somehow,
a standard fractal construction will result?
- If so, what should the w[k] be? (1/s[k]??), *and how do you conclude this?*

Thanks for any guidance (or discussion!)
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