2019
Том 71
№ 11

# Korneichuk N. P.

Articles: 30
Article (Russian)

### Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 579-594

Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ for all $x \in L^{r\gamma}_{\infty}(T^d)$ Moreover, if $\bar \beta$ is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$

Anniversaries (Ukrainian)

### Igor Volodymyrovych Skrypnik (On His 60th Birthday)

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1443-1445

Article (Russian)

### A brief survey of scientific results of E. A. Storozhenko

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 463-473

We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.

Article (Russian)

### On the best approximation of periodic functions of two variables by polynomial splines

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 52-57

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.

Article (Russian)

### Inequalities for polynomial splines

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 58-65

We establish exact estimates for the variation on a period of the derivative s (r)(t) of a periodic polynomial spline s(t) of degree r and defect 1 with respect to a fixed partition of [0, 2π) under the condition that $\left\| {s^{(r)} } \right\|_X = 1$ , where X=C or L 1

Article (Russian)

### Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 66-84

We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes $W^r H^{ω}$, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class $H^{ω}$ on the derivatives of trigonometric polynomials or polynomial splines in terms of the $L^{ϱ}$ -norms of these polynomials and splines.

Article (Ukrainian)

### On the best approximation of functions of $n$ variables

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1352–1359

We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions.

Article (Russian)

### Information aspects in the theory of approximation and recovery of operators

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 314–327

We present a brief review of new directions in the theory of approximation which are associated with the information approach to the problems of optimum recovery of mathematical objects on the basis of discrete information. Within the framework of this approach, we formulate three problems of recovery of operators and their values. In the case of integral operator, we obtain some estimates of the error.

Article (Russian)

### Permutations and piecewise-constant approximation of continuous functions of n variables

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 907–918

We consider the problem of approximation of a continuous function f given on an n-dirnensional cube by step functions in the metrics of C and L p. We obtain exact error estimates in terms of the modulus of continuity of the function f or a special permutation of it.

Anniversaries (Ukrainian)

### Anatolii Mikhailovich Samoilenko (on his 60th birthday)

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4

Article (Russian)

### Complexity of approximation problems

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1683-1694

We consider some aspects of optimal encoding and renewal related to the problem of complexity of the ε-definition of functions posed by Kolmogorov in 1962. We present some estimates for the ε-complexity of the problem of renewal of functions in the uniform metric and Hausdorff metric.

Article (Russian)

### On linear widths of classes $H^ω$

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1255-1264

We obtain new results related to the estimation of the linear widths $λ_N$ and $λ^N$ in the spaces $C$ and $L_p$ for the classes $H^ω$ (in particular, for $H^α,\; 0 < α < 1$).

Article (Russian)

### Information widths

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1506–1518

We introduce the notions of adaptive information widths of a set in a metric space and consider the problem of comparing them with nonadaptive widths. Exact results are obtained for one class of continuous functions that is not centrally symmetric.

Article (Russian)

### On the optimal reconstruction of the values of operators

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1375–1381

For a continuous operator $А:\; X \rightarrow Y,$ we formulate the problem of the optimal renewal of values $Аx,\; x \in X$ by decreasing the uncertainty domain by using an information $\mu_k(x),\; k = 1, 2, ...,$, where $\mu_k$ are continuous functionals, defined on the space $X$. Specific results are obtained for some integral operators in functional spaces.

Article (Ukrainian)

### Informativeness of functionals

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1156–1163

We introduce the concept of informativeness of a continuous functional on a metric spaceX with respect to a setM?X and a metric ?x. We pose the problem of finding the most informative functional. For some sets of continuous functions, this problem is solved by reduction to a subset of functionals given by the value of a function at a certain point.

Article (Russian)

### Optimization of adaptive algorithms for the renewal of monotone functions from the class $H^ω$

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1627–1634

A problem of renewal of monotone functions $f(t) \in H^{\omega}[a, b]$ with fixed values at the ends of an interval is studied by using adaptive algorithms for calculating the values of $f(t)$ at certain points. Asymptotically exact estimates unimprovable on the entire set of adaptive algorithms are obtained for the least possible number $N(\varepsilon)$ of steps providing the uniform $ε$-error. For moduli of continuity of type $εα, 0 < α < 1$, the value $N(\varepsilon)$ has a higher order as $ε → 0$ than in the nonadaptive case for the same amount of information.

Article (Russian)

### On passive and active algorithms of reconstruction of functions

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 258–264

We consider passive and active algorithms of reconstruction of functions, satisfying the condition $|f(t′) − f(t″)| ≤ |t′ − t″|^{α},\; 0 < α ≤ 1,$ according to their values $f(t)$ at the points of the interval $[a, b]$. An active algorithm is presented which guarantees, for monotonic functions from the above-mentioned class with $0 < α < 1$, a higher order of error in $C [a, b]$ than can be attained by any passive algorithm.

Article (Ukrainian)

### Some problems of coding and reconstructing functions

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 514-524

Article (Ukrainian)

### A derivation of exact estimates for the derivative of the spline-interpolation error

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 206-210

Article (Ukrainian)

### Behavior of the derivatives of the error of a spline interpolation

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 67–72

Article (Ukrainian)

### Approximation theory and optimization problems

Ukr. Mat. Zh. - 1990. - 42, № 5. - pp. 579–593

Article (Ukrainian)

### Optimal coding of vector-functions

Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 737-743

Article (Ukrainian)

### Optimal coding of elements of a metric space

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 168–173

Article (Ukrainian)

### Approximation of differential functions and their derivatives by parabolic splines

Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 702-710

Article (Ukrainian)

Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 804–805

Article (Ukrainian)

### Approximations by local splines of minimal defect

Ukr. Mat. Zh. - 1982. - 34, № 5. - pp. 617—621

Article (Ukrainian)

### Error bound of spline interpolation in an integral metric

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 391–394

Article (Ukrainian)

### Inequalities for best spline approximation of periodic differentiable functions

Ukr. Mat. Zh. - 1979. - 31, № 4. - pp. 380–388

Article (Ukrainian)

### On the approximation of continuous functions by algebraic polynomials

Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 326—339

Brief Communications (Russian)

### On the asymptotic estimate of the remainder in approximating periodic functions satisfying Lipshitz's condition by interpolation polynomials with equidistant nodes

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 100-106