Hmmm ... unless Σ is just a set of isolated points, the values on Σ actually determine the rest, don't they? In the sense, taking the difference of two functions of one complex variable, that if we have a function that vanishes on Σ it will just be zero everywhere. When Σ is finite, that's not trure, but it is also not much of a problem to have the functional calculuas for finite spectra?