Accession Number : ADA141606

Title :   A Geometric Proof of Total Positivity for Spline Interpolation.

Descriptive Note : Technical summary rept.,


Personal Author(s) : Boor,C de ; DeVore,R

PDF Url : ADA141606

Report Date : Apr 1984

Pagination or Media Count : 14

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.

Descriptors :   *Splines(Geometry), *Interpolation, Matrices(Mathematics), Coefficients, Polygons, Variations, Computer aided design

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE