A series related to the Hilbert transform

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

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/(2n-1) if n > 0
f(n) = - f(1-n) 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
square-summable sequences.

Google