A Geometric Proof of Total Positivity for Spline Interpolation.
Abstract
The total positivity of the spline collocation matrix is the basis of several important results in univariate spline theory. This makes it desirable to provide as simple as possible a proof of this total positivity. The proofs available in the literature don't qualify since these all rely on certain determinant identities which are not exactly intuitive. The authors give a proof that uses nothing more than Cramer's rule (hard to avoid since total positivity is a statement about determinants) and the geometrically obvious fact that a B-spline can always be written as a positive combination of B-splines on a finer knot sequence. The geometric intuition appealed to here stems from the area of Computer-Aided Design in which a spline is constructed and manipulated through its B-polygon, a broken line whose vertices correspond to the B-spline coefficients. If a knot is added (to provide greater potential flexibility), the new B-polygon is obtained by interpolation to the old. This had led Lane and Riesenfeld to a proof of the variation diminishing property of the spline collocation matrix and is shown here to provide a proof of the total positivity as well.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1984
- Accession Number
- ADA141606
Entities
People
- C. De Boor
- R. Devore
Organizations
- University of Wisconsin–Madison