A geometric proof of total positivity for spline interpolation

Authors:
C. de Boor and R. DeVore

Journal:
Math. Comp. **45** (1985), 497-504

MSC:
Primary 41A15; Secondary 65D07

MathSciNet review:
804938

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**W. Böhm, "Inserting new knots into B-spline curves,"*Computer-Aided Design*, v. 12, 1980, pp. 199-201.**[2]**Carl de Boor,*Total positivity of the spline collocation matrix*, Indiana Univ. Math. J.**25**(1976), no. 6, 541–551. MR**0415138****[3]**Carl de Boor,*A practical guide to splines*, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR**507062****[4]**Rong Qing Jia,*Total positivity of the discrete spline collocation matrix*, J. Approx. Theory**39**(1983), no. 1, 11–23. MR**713358**, 10.1016/0021-9045(83)90065-5**[5]**Samuel Karlin,*Total positivity. Vol. I*, Stanford University Press, Stanford, Calif, 1968. MR**0230102****[6]**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**, 10.1016/0021-9045(83)90111-9**[7]**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**0053177**, 10.1090/S0002-9947-1953-0053177-X

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804938-2

Keywords:
Spline interpolation,
total positivity,
variation diminishing

Article copyright:
© Copyright 1985
American Mathematical Society