Optimal analytic extrapolation for the scattering amplitude from cuts to interior points

Given a data function together with an error corridor for the scattering amplitude along some finite part of the cuts, one can construct effectively the whole set of analytic functions (``admissible amplitudes''), compatible with these conditions and bounded by a certain number M on the remaining part of the cuts. Depending on the actual value of an important constant ε₀ computed from the data function and the bound M, this set may be void. If not, in every point of the cut plane the set of values of the admissible amplitudes fills densely a circle; explicit formulas are given for its radius $\mathrm{\eta ̂(z)}$ and center f̂(z), the latter being the best possible estimate for the whole set. In contrast to the linear extrapolation obtained by Poisson weighted dispersion relations, here nonlinear functional methods were used. This paper contains an appendix written by Professor C. Foias, on some functional analytical methods used in connection with the computation of the numerical value of the constant ε₀.