Total positivity properties of LB-splines

Abstract. A basic property of both polynomial and more general Tchebycheffian splines is that the associated B-spline collocation matrix has certain total positivity properties, and it is nonsingular if and only if well-known interlacing conditions hold. Using a technique from a recent paper of Mørken, we extend these results to L-splines. One of the most important properties of classical polynomial B-splines is the fact that matrices formed from ordered subsets of them by evaluation at a set of (possibly repeated) points always have nonnegative determinants which are positive if and only if the knots and interpolation points interlac

On Factorizations of Totally Positive Matrices

Abstract. Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given. Totally positive matrices (TP matrices in the sequel) are real, nonnegative matrices whose all minors are nonnegative. They have a long history and many applications (see the paper by Allan Pinkus in this volume for the early history and motivations) and have been studied mainly by researchers of thos