A Geometric Proof of Total Positivity for Spline Interpolation.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
Pagination or Media Count:
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 dont qualify since these all rely on certain determinant identities which are not exactly intuitive. The authors give a proof that uses nothing more than Cramers 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.
- Theoretical Mathematics