##
**Shape preserving representations and optimality of the Bernstein basis.**
*(English)*
Zbl 0832.41013

The paper deals with shape preserving properties of totally positive (TP) bases of polynomials of degree less than or equal to \(n\).

A sequence of functions \(U:= (u_0, \dots, u_n)\) is TP on an interval \(I\) if for any \(t_0< t_1< \dots< t_m\) in \(I\) the collocation matrix \((u_j (t_i) )_{i=0, \dots, m, j=0, \dots, n}\) is TP, i.e., if all its minors are nonnegative. Given a sequence of positive functions \(U:= (u_0, \dots, u_n)\) on \([a, b]\) with \(\sum_{i=0}^n u_i (t)=1\) and a sequence \((C_0, \dots, C_n)\) of points in \(\mathbb{R}^k\), let \[ \gamma(t)= \sum_{i=0}^n C_i u_i (t) \qquad (t\in [a, b]). \] If \(U\) is TP, then the curve \(\gamma\) preserves many shape preserving properties of the control polygon generated by \((C_0, \dots, C_n)\).

In the paper the following conjecture of T. N. Goodman and H. B. Said [Comput. Aided Geom. Des. 8, No. 2, 115-121 (1991; Zbl 0729.65006)] is proved: The Bernstein basis has optimal shape preserving properties among all normalized TB bases of polynomials of degree less than or equal to \(n\) over a compact interval.

Further, a simple test for recognizing normalized totally positive bases with good shape preserving properties is proposed. Moreover, a corresponding corner cutting algorithm to generate the Bézier polygon is derived.

A sequence of functions \(U:= (u_0, \dots, u_n)\) is TP on an interval \(I\) if for any \(t_0< t_1< \dots< t_m\) in \(I\) the collocation matrix \((u_j (t_i) )_{i=0, \dots, m, j=0, \dots, n}\) is TP, i.e., if all its minors are nonnegative. Given a sequence of positive functions \(U:= (u_0, \dots, u_n)\) on \([a, b]\) with \(\sum_{i=0}^n u_i (t)=1\) and a sequence \((C_0, \dots, C_n)\) of points in \(\mathbb{R}^k\), let \[ \gamma(t)= \sum_{i=0}^n C_i u_i (t) \qquad (t\in [a, b]). \] If \(U\) is TP, then the curve \(\gamma\) preserves many shape preserving properties of the control polygon generated by \((C_0, \dots, C_n)\).

In the paper the following conjecture of T. N. Goodman and H. B. Said [Comput. Aided Geom. Des. 8, No. 2, 115-121 (1991; Zbl 0729.65006)] is proved: The Bernstein basis has optimal shape preserving properties among all normalized TB bases of polynomials of degree less than or equal to \(n\) over a compact interval.

Further, a simple test for recognizing normalized totally positive bases with good shape preserving properties is proposed. Moreover, a corresponding corner cutting algorithm to generate the Bézier polygon is derived.

Reviewer: G.Plonka (Rostock)

### MSC:

41A36 | Approximation by positive operators |

41A10 | Approximation by polynomials |

65D17 | Computer-aided design (modeling of curves and surfaces) |

### Citations:

Zbl 0729.65006
PDF
BibTeX
XML
Cite

\textit{J. M. Carnicer} and \textit{J. M. Peña}, Adv. Comput. Math. 1, No. 2, 173--196 (1993; Zbl 0832.41013)

Full Text:
DOI

### References:

[1] | T. Ando, Totally positive matrices, Lin. Alg. Appl. 90(1987)165–219. · Zbl 0613.15014 |

[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. · Zbl 0479.15015 |

[3] | C.W. Cryer, The LU-factorization of totally positive matrices, Lin. Alg. Appl. 7(1973)83–92. · Zbl 0274.15004 |

[4] | M. Gasca and J.M. Peña, Total positivity and Neville elimination, Lin. Alg. Appl. 165(1992)25–44. · Zbl 0749.15010 |

[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. · Zbl 0758.15014 |

[6] | M. Gasca and J.M. Peña, Total positivity, QR factorization and Neville elimination, SIAM J. Matrix Anal. (1992), to appear. · Zbl 0749.15010 |

[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. |

[8] | T.N.T. Goodman, Inflections on curves in two and three dimensions, Comput. Aided Geom. Design 8(1991)37–50. · Zbl 0719.65011 |

[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. · Zbl 0652.41003 |

[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. · Zbl 0729.65006 |

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

[12] | C.A. Micchelli and A. Pinkus, Descartes systems from corner cutting, Con. Approx. 7(1991)161–194. · Zbl 0784.65017 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.