The Linear, Functional Equation Approach to the / Problem of the Convergence of Pads Approximants*

; The Pade approximant problem is related to a (not necessarily orthogonal) projection of a linear functional equation of the Fredholni type. If the kernel is of trace class and its upper Hessenberg form is tridiagonal (this class includes Hermitian operators), then we prove that not only do the diagonal Pade' appioximants converge, but so do their numerators and denominators separately. The generalization of these results to C classes of compact operators is given. For kernels P 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. *Work performed under the auspices of the U.S. ERDA. Considerable progress in the study of the convergence of Pade' approximants can be made, I think, by the use of the techniques of functional equations. What I will report here is probably just a beginning, and is drawn in part from a previous paper.Cl] First we will review the known relation of Pade' approxiinants to linear functional equations, then we review the properties of some special classes of compact operator, and give convergence results for these classes. Finally we indicate how these results lead to convergence of Pade approxiinants to the partial wave scattering amplitudes in certain quantun 1 mechanical scattering problems. PROJECTIONS IH THE CINI-FUBINI SUBSPACE Suppose we consider the functional equation f • g + X A f (1) where f, g, and h belong to some Hilbert space W, and A is a linear operator whose properties are yet to be defined. We also introduce the associated sets of elements cpA = A*" 1 g , <p̂ . (A ) " 1 h i « 1, 2 (2) where A i« the Hermitian conjugate operator to A. We need as well the N x N matrix i,j " ( V V = ' " 9> w defined in terms of the inner products of the cp.. and tp.. We are now in a position to define our projection operator onto the Cini-Fubini subspace [2] N -1 P E q> (R V <cp,. ) , (4) M x *3 J provided det|RJ ^ 0. (It can be shown [3] that there exists an infinite number of such N's.) The operator P is a projection on 3 from S... (The spaces »„ and S., are respectively those spaces spanned by cp. and cp. for i = 1,...,N.) It has the properties