Coconvex Approximation of Functions with More than One Inflection Point - Dzyubenko, H. A.; Zalizko, V. D. (Ukrains’kyi Matematychnyi Zhurnal, 2004)
Assume that f ∈ C[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y i, \(1,s \) , from (−1, 1). For n ∈ N, n ≥ 2, we construct an algebraic polynomial P n of order ≤ n that changes its convexity at the same points y i as f and is such that |f(x) - P_n (x)| C(Y)ω _3 ( f;1n^2 + 1 - x^2 n ), x [ - 1, 1], where ω3(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on \(_i = 0,...,s | y_i - y_i + 1 |, y_0 = 1, y_s + 1 = - 1\) , y 0 = 1, y s + 1 = −1.