You can see that a lot of terms cancel. So, we can also change the type of problem and work backwards, setting the spectrum and working backwards to find the corresponding impulse response, and interpolation polynomial.
Chuck, thanks for the explanation! One question about this -- would working back from the spectrum possibly mean that you wouldn't necessarily match any of the points (would this mean you could say the impulse response would not have a value of 1 at t=0)? This would be an interesting proposition, and maybe it might help temper the tendency of polynomial interpolation to deliver values >1 or <-1 when the curve has a maximum or minimum between samples? I'm not sure if that would be the result, but I like the sentiment. =o)
Anyway, thanks again.
Matt