2019
Том 71
№ 4

Leviatan D.

Articles: 3
Article (English)

On one estimate of divided differences and its applications

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 230-245

We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their generalization to the Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f \in C(r)(I)$ and a set $Z = \{ z_j\}^{\mu}_{j=0}$ such that $z_{j+1} - z_j \geq \lambda | I|$ for all $0 \leq j \leq \mu 1$, where $I := [z_0, z_{\mu} ], | I|$ is the length of $I$, and $\lambda$ is a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality $\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$, $$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1} \int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$ where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj |$.

Article (English)

On moduli of smoothness with Jacobi weights

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 379-403

We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty$. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent to certain “Realization functionals”.

Article (English)

Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 369–386

In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters.