2019
Том 71
№ 7

# Zalizko V. D.

Articles: 3
Article (Ukrainian)

### Coconvex approximation of periodic functions

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 29–43

The Jackson inequality E n (f ) ≤ c ω 3 (f , π / n ) connects the value of the best uniform approximation E n (f ) of a 2π-periodic function f : RR by trigonometric polynomials of order ≤ n — 1 with its third modulus of continuity ω 3 (f, t ).
In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity on [—π, π) only at every point of a fixed finite set consisting of the even number of points are approximated by polynomials coconvex to them.

Article (Ukrainian)

### Pointwise Estimates for the Coconvex Approximation of Differentiable Functions

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 47–59

We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$.

Article (Ukrainian)

### Coconvex Approximation of Functions with More than One Inflection Point

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 352-365

Assume that fC[−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 nN, 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.