OPTIMAL APPROXIMATION BY “ALMOST CLASSICAL” INTERPOLATION

Abstract

Optimal approximation in a Szego-Hilbert space is shown to be equivalent to the problem of constructing a certain rational interpolating function. This interpolation problem leads to techniques analogous to those developed for classical polynomial interpolation; in particular, forms equivalent to the Lagrange polynomial, the Neville-Aitken method and the Newton polynomial constructed from generalized divided differences, are exhibited. Since polynomial interpolation obtains in the limit as the domain of the functions in the Szego-Hilbert space becomes infinite, the classical dichotomy of polynomial interpolation and approximation of functions is unified.