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Pointwise estimation of comonotone approximation

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Abstract

We prove that, for a continuous functionf(x) defined on the interval [−1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)−P n (x) ¦≤Cω2(f;n−2+n −1√1−x 2), whereC is a constant that depends on the length of the smallest interval.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    G. A. Dzyubenko

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  1. G. A. Dzyubenko
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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.

The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.

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Dzyubenko, G.A. Pointwise estimation of comonotone approximation. Ukr Math J 46, 1620–1626 (1994). https://doi.org/10.1007/BF01058880

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  • DOI: https://doi.org/10.1007/BF01058880

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