Stable extrapolation of scattering amplitudes using unitarity

Unitarity in all three channels for the pi N system is used to show that one can, given the first L pi N partial waves and the pi N total cross sections, extrapolate to find the NN to pi pi waves, the whole amplitude, and the pi N waves, in the unphysical regions and this within absolute error bands, and from either of those quantities, construct Pade approximants which are then shown to converge to the cuts (4 mu /sup 2/<or=t<or approximately=2M/sup 2/) again here within an explicit error band. It is shown that the method is stable under perturbations, giving also the explicit form in which such perturbations propagate. The analysis is carried first for amplitudes with positivity, and then extended to any amplitude. A general outline is presented. (31 refs).