Parfinovych N. V.

We obtain the exact values of the best $(\alpha , \beta )$-approximations of the classes $K \ast F$ of periodic functions $K \ast f$ such that $f$ belongs to a given rearrangement-invariant set $F$ and $K$ is $2\pi$ -periodic kernel that do not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points $2k\pi /n$ and $2k\pi /n + h, n \in N, k \in Z, h \in (0, 2\pi /n)$. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

We obtain new exact Landau-type estimates for the uniform norms of one-dimension Riesz potentials of the partial derivatives of a multivariable function in terms of the norm of the function itself and the norm of its Laplacian.

Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented.

In the case where $n → ∞$, we obtain order equalities for the best $L_q$ -approximations of the classes $W_p^r ,\; 1 ≤ q ≤ p ≤ 2$, of differentiable periodical functions by splines from these classes.

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.

As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions.

We obtain the exact asymptotics (as n → ∞) of the best L ₁-approximations of classes \(W_1^r\) of periodic functions by splines s ∈ S _{2n, r − 1} and s ∈ S _{2n, r + k − 1} (S _{2n, r} is the set of 2π-periodic polynomial splines of order r and defect 1 with nodes at the points kπ/n, k ∈ Z) under certain restrictions on their derivatives.

We find the exact asymptotics ($n → ∞$) of the best $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s ∈ S_{2n, r∼-1}$ ($S_{2n, r∼-1}$ is a set of $2π$-periodic polynomial splines of order $r−1$, defect one, and with nodes at the points $kπ/n,\; k ∈ ℤ$) such that $V_0^{2π} s^{( r-1)} ≤ 1+ɛ_n$, where $\{ɛ_n\}_{n=1}^{ ∞}$ is a decreasing sequence of positive numbers such that $ɛ_n n^2 → ∞$ and $ɛ_n → 0$ as $n → ∞$.