Suppose we have a time series $h(t)$ of length $n$. Let $H(f)$ be the Fourier transform of $h(t)$, i.e.
$$ H(f) = \sum_{t=1}^{n} h(t) e^{-i 2 \pi f t}$$
Now I want to: 1) compute $H(f+f')$, and 2) re-write $H(f+f')$ in terms of $H(f)$.
$$ H(f+f') = \sum_{t=1}^{n} h(t) e^{-i 2 \pi (f + f') t} = \sum_{t=1}^{n} h(t) e^{-i 2 \pi f t} e^{-i 2 \pi f' t} $$
Since I am summing over $t$, I can't find a way to simplify further. Any tips? Thanks!
I'm afraid that is as simple as it gets. But you can say that if $H(f)$ is the transform of $h(t)$, then the inverse transform of $H(f+f')$ is $h(t){e^{-i 2 \pi f' t}}$. Rearranging your equation slightly shows why this is so.
$$ H(f+f') = \sum_{t=1}^{n} \left[ h(t){e^{-i 2 \pi f' t}}\right] e^{-i 2 \pi f t} $$ This is one of the best books on the FFT.