Coconvex Pointwise Approximation - Dzyubenko, G.A.; Gilewicz, J.; Shevchuk, I.A. (Український математичний журнал, 2002)
Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points yi ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial Pn of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points yi as f and |f(x)−Pn(x)| ≤ cω₂ (f, (√(1−x²))/n,x∈[−1,1], 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.