Abstract
Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and
where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.
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Dzyubenko, G.A., Gilewicz, J. & Shevchuk, I.A. Coconvex Pointwise Approximation. Ukrainian Mathematical Journal 54, 1445–1461 (2002). https://doi.org/10.1023/A:1023411817844
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DOI: https://doi.org/10.1023/A:1023411817844