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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.


References [Enhancements On Off] (What's this?)

  • [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

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