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Negative result in pointwise 3-convex polynomial approximation

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Ukrainian Mathematical Journal Aims and scope

Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function fC r [−1, 1] ⋂ Δ3 [−1, 1] such that ∥f (r) C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ3 [−1, 1], there exists x such that

$$ \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), $$

where C > 0 is a constant that depends only on r, and

$$ {{{\uprho }}_n}(x): = \frac{1}{{{n^2}}} + \frac{1}{n}\sqrt {1 - {x^2}} . $$

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 4, pp. 563–567, April, 2009.

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Bondarenko, A.V., Gilewicz, J.J. Negative result in pointwise 3-convex polynomial approximation. Ukr Math J 61, 674–681 (2009). https://doi.org/10.1007/s11253-009-0228-7

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  • DOI: https://doi.org/10.1007/s11253-009-0228-7

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