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

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

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]**C. de Boor, "Total positivity of the spline collocation matrix,"*Indiana Univ. Math. J.*, v. 25, 1976, pp. 541-551. MR**0415138 (54:3229)****[3]**C. de Boor,*A Practical Guide to Splines*, Applied Mathematical Sciences, Vol. 27, Springer-Verlag, Berlin and New York, 1978. MR**507062 (80a:65027)****[4]**Jia Rong-Qing, "Total positivity of the discrete spline collocation matrix",*J. Approx. Theory*, v. 39, 1983, pp. 11-23. MR**713358 (84h:41016)****[5]**S. Karlin,*Total Positivity*, Vol. I, Stanford Univ. Press, Stanford, Calif., 1968. MR**0230102 (37:5667)****[6]**J. Lane & R. Riesenfeld, "A geometric proof for the variation diminishing property of B-spline approximation,"*J. Approx. Theory*, v. 37, 1983, pp. 1-4. MR**685351 (84g:41006)****[7]**I. J. Schoenberg & A. Whitney, "On Pólya frequency functions. III,"*Trans. Amer. Math. Soc.*, v. 74, 1953, pp. 246-259. MR**0053177 (14:732g)**

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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