Linear, functional equation approach to the problem of the convergence of Pade approximants

The Pade approximant problem is related to a (not necessarily orthogonal) projection of a linear functional equation of the Fredholm type. If the kernel is of trace class and its upper Hessenberg form is tridiagonal (this class includes Hermitian operators), then it is proven that not only do the diagonal Pade approximants converge, but so do their numerators and denominators separately. The generalization of these results to C/sub p/ classes of compact operators is given. For kernels which are not only compact, but also satisfy an additional mild restriction, a pointwise convergence theorem is proven. The application of these results to quantum scattering theory is indicated. (auth)