# Negative result in pointwise 3-convex polynomial approximation

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Let Δwhere

^{3}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] ⋂ Δ^{3}[−1, 1] such that ∥*f*^{(r)}∥_{ C[−1, 1]}≤ 1 and, for an arbitrary algebraic polynomial*P*∈ Δ^{3}[−1, 1], there exists*x*such that$$ \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), $$

*C*> 0 is a constant that depends only on*r*, and$$ {{{\uprho }}_n}(x): = \frac{1}{{{n^2}}} + \frac{1}{n}\sqrt {1 - {x^2}} . $$

## Keywords

Function Convex Naukova Dumka Polynomial Approximation Uniform Approximation Lagrange Form
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## References

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© Springer Science+Business Media, Inc. 2009