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generalized bernstein polynomial

Proceedings of the Edinburgh Mathematical Society (1)

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The r-monotonicity of generalized Bernstein polynomials
Part of
- Approximations and expansions
Laiyi Zhu, Zhiyong Huang
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 55 / Issue 3 / October 2012

Published online by Cambridge University Press:

26 July 2012, pp. 797-807
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Let f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.