  We generalize a recent model-independent form factor parameterization derived
from rigorous dispersion relations to include constraints from data in the
timelike region. These constraints dictate the convergence properties of the
parameterization and appear as sum rules on the parameters. We further develop
a new parameterization that takes into account finiteness and asymptotic
conditions on the form factor, and use it to fit to the elastic \pi
electromagnetic form factor. We find that the existing world sample of timelike
data gives only loose bounds on the form factor in the spacelike region, but
explain how the acquisition of additional timelike data or fits to other form
factors are expected to give much better results. The same parameterization is
seen to fit spacelike data extremely well.
