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**Convexity and generalized Bernstein polynomials.**
*(English)*
Zbl 0930.41010

Authors’ abstract. In a recent generalization of the Bernstein polynomials, the approximated function \(f\) is evaluated at points spaced at intervals which are in geometric progression on \([0,1]\), instead of equally spaced points. For each positive integer \(n\), this replaces the single polynomial \(B_nf\) by a one-parameter family of polynomials \(B^q_nf\), where \(0\leq q\leq 1\). This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and give new results concerning \(B_n^qf\) when \(f\) is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if \(f\) is increasing then \(B_n^qf\) is increasing, and if \(f\) is convex then \(B^q_nf\) is convex, generalizing well known results when \(q=1\). It is also shown that if \(f\) is convex, then for any positive integer \(n\), \(B_n^rf\leq B^q_nf\) for \(0<q\leq r\leq 1\). This supplements the well known classical result that \(f\leq B_nf\) when \(f\) is convex.

Reviewer: E.Deeba (Houston)

### MSC:

41A10 | Approximation by polynomials |

### Keywords:

Bernstein polynomials
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\textit{T. N. T. Goodman} and \textit{G. M. Phillips}, Proc. Edinb. Math. Soc., II. Ser. 42, No. 1, 179--190 (1999; Zbl 0930.41010)

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### References:

[1] | Schoenberg, Proc. Roy. Soc. Edinburgh 90A pp 195– (1981) · Zbl 0506.41003 |

[2] | Phillips, BIT 36 pp 232– (1996) |

[3] | Goodman, Total Positivity and its Applications pp 157– (1996) |

[4] | Lee, Proc. Roy. Soc. Edinburgh 108A pp 75– (1988) · Zbl 0639.65006 |

[5] | Phillips, Ann. Numer. Math. 4 pp 511– (1997) |

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