Autocorrelation
- From: "Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx>
- Date: Sat, 8 Oct 2005 07:41:32 -0500
I was looking at the sum
sum(sum(1/k^s*1/(m+k)^s,k=1..oo),m=1..oo)
and noticed it looks a lot like the total of the autocorrelation of 1/k^s
i.e.
if we take the autocorrelation of 1/k^s we get
c(m) = sum(1/k^s/(k+m)^s,k=1..oo)
and hence the sum above is the "total correlation"
but we know that there is a relation between the fourier series and the
autocorrelation given at
http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html
hence
c*(m) = F_m[1/k^(2s)]
and we "should" be able to compute the original sum by summing over c* from
1 to oo.
I didn't get anywhere but seems like the idea should be correct(asside from
the technicalities that I didn't really get into...)?
Jon
.
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