Negative result in pointwise 3-convex polynomial approximation |
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| Authors: | A V Bondarenko J J Gilewicz |
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| Institution: | 1.Shevchenko Kyiv National University,Kyiv,Ukraine;2.Centre de Physique Théorique,Marseille,France |
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| Abstract: | 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 f ∈ C
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
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| f(x) - P(x) | 3 C?n \uprhonr(x), \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), |
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| Keywords: | |
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