2019
Том 71
№ 11

# Shchitov A. N.

Articles: 2
Article (Russian)

### On some extremal problems in the theory of approximation of functions in the spaces $S^p,\quad 1 \leq p < \infty$

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 303-316

We consider and study properties of the smoothness characteristics $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$, of functions $f(x)$ that belong to the space $S^p,\quad 1 \leq p < \infty$, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using $\Omega_m(f, t)_{S^p},\quad m \in \mathbb{N},\quad t > 0$ are calculated.

Article (Russian)

### Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1458-1466

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions $f(x) ∈ L_2^r(r ∈ ℤ_{+})$ and expressions containing moduli of continuity of the $k$th order $ω_k(f^{(r)}, t)$. Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from $L_2$. For the classes $F (k, r, Ψ*)$ defined by $ω_k(f^{(r)}, t)$ and the majorant $Ψ(t)=t^{4k/π^2}$, we determine the exact values of different widths in the space $L_2$.