I would like to show that the following summation is establishes an ${{{{{}}}}}$orthonormal relation:
$\displaystyle \delta_{mk}= \frac{1}{2N}\sum_{p=0}^{2N-1} \exp(i\omega_p (t_m - t_k)) : \omega_p := \frac{2\pi p}{T} , t_p := \frac{pT}{2N} : N, p,k \in \mathbb{N} , T>0$
What I have:
the ratio between subsequent terms is
$r=\exp(\frac{2 \pi i }{T})$ so I am motivated by adding partial sums (in particular the "total" (partial) sum): $\frac{1-\exp(2\pi i /T)^{2N}}{1-\exp(2\pi i /T)}$
I see very clearly that if $m=k$ the series sums to 1 (after normalizing), this is obvious. I could use some help on the other half. Thanks!
