A geometric proof of total positivity for spline interpolation

de Boor, C.; DeVore, R.

doi:10.1090/S0025-5718-1985-0804938-2

A geometric proof of total positivity for spline interpolation
HTML articles powered by AMS MathViewer

by C. de Boor and R. DeVore PDF

Math. Comp. 45 (1985), 497-504 Request permission

Abstract:

Simple geometric proofs are given for the total positivity of the B-spline collocation matrix and the variation diminishing property of the B-spline representation of a spline.

References

Computer-Aided Design

Carl de Boor, Total positivity of the spline collocation matrix, Indiana Univ. Math. J. 25 (1976), no. 6, 541–551. MR 415138, DOI 10.1512/iumj.1976.25.25043
Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
Rong Qing Jia, Total positivity of the discrete spline collocation matrix, J. Approx. Theory 39 (1983), no. 1, 11–23. MR 713358, DOI 10.1016/0021-9045(83)90065-5
Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968. MR 0230102
Jeffrey M. Lane and Richard F. Riesenfeld, A geometric proof for the variation diminishing property of $B$-spline approximation, J. Approx. Theory 37 (1983), no. 1, 1–4. MR 685351, DOI 10.1016/0021-9045(83)90111-9
I. J. Schoenberg and Anne Whitney, On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc. 74 (1953), 246–259. MR 53177, DOI 10.1090/S0002-9947-1953-0053177-X