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Shape preserving representations and optimality of the Bernstein basis

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Abstract

This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ∞) is also totally positive.

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References

  1. T. Ando, Totally positive matrices, Lin. Alg. Appl. 90(1987)165–219.

    Google Scholar 

  2. C. de Boor and A. Pinkus, The approximation of a totally positive matrix by a strictly banded totally positive one, Lin. Alg. Appl. 42(1982)81–98.

    Google Scholar 

  3. C.W. Cryer, The LU-factorization of totally positive matrices, Lin. Alg. Appl. 7(1973)83–92.

    Google Scholar 

  4. M. Gasca and J.M. Peña, Total positivity and Neville elimination, Lin. Alg. Appl. 165(1992)25–44.

    Google Scholar 

  5. M. Gasca and J.M. Peña, On the characterization of TP and STP matrices, to appear in:Approximation Theory, Spline Functions and Applications, ed. S.P. Singh (Kluwer Academic, Dordrecht, 1992) pp. 357–364.

    Google Scholar 

  6. M. Gasca and J.M. Peña, Total positivity, QR factorization and Neville elimination, SIAM J. Matrix Anal. (1992), to appear.

  7. T.N.T. Goodman, Shape preserving representations, inMathematical Methods in CAGD, ed. T. Lyche and L.L. Shumaker (Academic Press, Boston, 1989) pp. 333–357.

    Google Scholar 

  8. T.N.T. Goodman, Inflections on curves in two and three dimensions, Comput. Aided Geom. Design 8(1991)37–50.

    Google Scholar 

  9. T.N.T. Goodman and C.A. Micchelli, Corner cutting algorithms for the Bézier representation of free form curves, Lin. Alg. Appl. 99(1988)225–252.

    Google Scholar 

  10. T.N.T. Goodman and H.B. Said, Shape preserving properties of the generalized Ball basis, Comput. Aided Geom. Design 8(1991)115–121.

    Google Scholar 

  11. S. Karlin,Total Positivity (Stanford University Press, Stanford, 1968).

    Google Scholar 

  12. C.A. Micchelli and A. Pinkus, Descartes systems from corner cutting, Con. Approx. 7(1991)161–194.

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Zaragoza, Zaragoza, Spain

    J. M. Carnicer & J. M. Peña

Authors
  1. J. M. Carnicer
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  2. J. M. Peña
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Both authors were partially supported by DGICYT PS90-0121.

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Carnicer, J.M., Peña, J.M. Shape preserving representations and optimality of the Bernstein basis. Adv Comput Math 1, 173–196 (1993). https://doi.org/10.1007/BF02071384

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  • DOI: https://doi.org/10.1007/BF02071384

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