Nonpolynomial interpolation

Abstract

Here interpolation is meant in the following sense: given f ε C¦a, b¦. and given a set of distinct points in ¦ a, b¦ and a linearly independent set {ifu 0,...,u n} of continuous functions on ¦ a, b¦, the interpolating function L n f is the unique linear combination of u 0,..., u n that coincides with f at the given points, if such a linear combination exists. In the classical case of Lagrange interpolation. u i ( x) is a polynomial of degree i. Here we allow other choices, and prove a generalization of the mean-convergence theorem of Erdös and Turán: it is shown that if a certain condition is satisfied, then L n f converges to f, in an appropriate L 2 sense, for all continuous functions f for which E n(f) →0 where E n(f) is the error of best uniform approximation by a linear combination of u 0,..., u n . In particular, this mean convergence property is shown to hold for interpolation by the leading eigenfunctions of a regular Sturm-Liouville eigenvalue problem, if the interpolation points are taken to be the zeros of the “next” eigenfunction. (The cigenfunctions are ordered so that the eigenvalues increase.)