2019
Том 71
№ 7

All Issues

Zalizko V. D.

Articles: 3
Article (Ukrainian)

Coconvex approximation of periodic functions

Zalizko V. D.

↓ Abstract   |   Full text (.pdf)

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

Dzyubenko H. A., Zalizko V. D.

↓ Abstract   |   Full text (.pdf)

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

Dzyubenko H. A., Zalizko V. D.

↓ Abstract   |   Full text (.pdf)

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.