Coconvex Approximation of Functions with More than One Inflection Point

Abstract

Assume that f ∈ C[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y _i, $\overline {1,s}$, 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

$$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$

where ω₃(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on $\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1$, y ₀ = 1, y _{s + 1} = −1.