Totally Positive Systems and Interpolation by Subspaces of Generalized Splines

Abstract

A subclass of generalized spline spaces which admit a totally positive system of functions and satisfy an interlacing property for unique interpolation is introduced. Moreover, interpolation by more general subspaces of spline functions is studied. It is shown that every regular space of generalized splines is a weak Chebyshev space if and only if unique interpolation can be characterized by a condition of Schoenberg-Whitney type.