Some Poised and Nonpoised Problems of Interpolation

Abstract

Taylor’s formula may be looked upon as a one-point interpolation formula with remainder. In 1906, Birkhoff used this simple observation to make a unified approach to the various interpolation and quadrature formulas of numerical analysis. Later, in 1931, Pólya gave a simple criterion to determine whether a given two-point interpolation problem is uniquely solvable (or poised). Interest in these questions was revived recently in 1966 by I. J. Schoenberg who generalized the result of Pólya in one direction and also introduced the idea of using an incidence matrix to describe an interpolation problem. Earlier P. Táran had initiated the study of “lacunary” interpolation on specially chosen abscissas and had shown its relevance to problems of convergence and approximation. These problems of P. Táran have a unique solution only for some specially chosen abscissas and so are not poised. Here we give a survey of the recent progress in this field with some applications and open problems.