Convergence of Padé approximants using the solution of linear functional equations

Abstract

We prove that a projection of the solutions to a linear functional equation of the Fredholm type with a compact kernel, projected into the Cini−Fubini subspaces, converge strongly to the solution in the whole space. Here either the whole sequence converges for all nonsingular points of the functional equation with at most one exceptional point, or by selecting at most two infinite subsequences we can obtain convergence for all nonsingular points. We then prove that the diagonal Padé approximants to the inner product of the solution with another element converge. For certain kernels of trace class, the numerator and denominator separately converge. As applications of these results, we prove the pointwise convergence of the Padé approximants to a wide class of meromorphic functions. We also prove the convergence, for decent potentials, of the Padé approximants to the scattering amplitudes for nonrelativistic quantum mechanical scattering problems. The numerators and denominators of the Padé approximants to the partial wave scattering amplitudes for single signed potentials converge separately to entire