A Lie-theoretic setting for the classical interpolation theories

Abstract

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane.