Interpolation by Weak Chebyshev Spaces

We present two characterizations of Lagrange interpolation sets for weak Chebyshev spaces. The rst of them is valid for an arbitrary weak Chebyshev space U and is based on an analysis of the structure of zero sets of functions in U extending Stockenberg's theorem. The second one holds for all weak Chebyshev spaces that possess a locally linearly independent basis. x1. Introduction Let U denote a nite-dimensional subspace of real valued functions deened on a totally ordered set K, for example, an arbitrary subset of IR. A nite subset T = ft 1 ; : : : ; t n g of K, where n = dimU, is called an interpolation set (I-set) w.r.t. U if for any given data fy 1 ; : : : ; y n g there exists a unique function u 2 U such that It is easy to see that T is an I-set w.r.t. U if and only if dimU jT = n; where U jT := fu jT : u 2 Ug. For a set of s points, T = ft 1 ; : : : ; t s g K, with s < n, we say that T is an I-set if dimU jT = s. We are interested in describing I-sets w.r.t. U in the case when U is a weak Chebyshev space (W T-space), i.e., every u 2 U has at most n?1 sign changes. The primary example of a WT-space is the space of univariate polynomial splines, in which case all interpolation sets can be characterized by well-known Schoenberg-Whitney condition (see e.g. 12]). Extensions of Schoenberg-Whitney theorem to some classes of generalized spline spaces were proposed in 11,13,14]. Recently, some characterizations of I-sets w.r.t. weak Chebyshev spaces without any a priori assumption about \piecewise Haar" spline-like structure have been found. In 4] it was proved that Schoenberg-Whitney characterization in its \di-mension form" holds true for a WT-space U if and only if U jK 0 is also a WT-space for all K 0 K. This last property is satissed, for example, if U is a weak Descartes space. In 2] the \support form" of Schoenberg-Whitney theorem has been shown to hold true for every WT-space that possesses a locally linearly independent weak Descartes basis. (See 7] for a review of various forms of Schoenberg-Whitney condition , especially in regard to their extendibility to multivariate splines.)