Interpolation and Almost Interpolation by Weak Chebyshev Spaces

Some new results on univariate interpolation by weak Cheby-shev spaces, using conditions of Schoenberg-Whitney type and the concept of almost interpolation sets, are given. x1. Introduction Let U denote a nite-dimensional subspace of real-valued functions deened on some set K. We are interested in describing those conngurations T = dimU jT = s: T is called an interpolation set (I-set) w.r.t. U. If s = n, then it is clearly equivalent to the condition that for any given data fy 1 ; : : : ; y n g there exists a unique u 2 U such that It is well known that in the case of univariate polynomial spline spaces all interpolation sets can be characterized by the Schoenberg-Whitney condition (see, e.g., 3, 4]). A new approach to multivariate interpolation has been found by Som-mer and Strauss using the concept of almost interpolation. A set T = t 0 i 2 B i such that T 0 = ft 0 1 ; : : : ; t 0 s g is an I-set w.r.t. U. They have shown that for a wide class of generalized spline spaces deened on polyhedral partitions AI-sets can be characterized by conditions of Schoenberg-Whitney type (for detail see 3]). All rights of reproduction in any form reserved.