larryhammick@telus.net science forum Guru Wannabe
Joined: 14 Feb 2006
Posts: 217

Posted: Fri Jul 21, 2006 3:28 am Post subject:
A series related to the Hilbert transform



This thing is not new, but it struck me as surprising and interesting,
so I thought I'd pass it along.
Define a function f on the integers by
f(n) = 1/(2n1) if n > 0
f(n) =  f(1n) if n <= 0.
For any fixed integer m, the family of numbers
f(n)f(n+m)
is absolutely summable, and its sum is zero if m is nonzero.
This result is essentially a special case of the fact that the
"discrete Hilbert transform" (google) preserves orthogonality between
squaresummable sequences. 
