Korneichuk N. P.
Kolmogorov-type inequalities for mixed derivatives of functions of many variables
Babenko V. F., Korneichuk N. P., Pichugov S. A.
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 $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$
Igor Volodymyrovych Skrypnik (On His 60th Birthday)
Berezansky Yu. M., Kharlamov P. V., Khruslov E. Ya., Kit G. S., Korneichuk N. P., Korolyuk V. S., Kovalev A. M., Kovalevskii A. A., Lukovsky I. O., Mitropolskiy Yu. A., Samoilenko A. M., Savchenko O. Ya.
Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1443-1445
A brief survey of scientific results of E. A. Storozhenko
Kashin B. S., Korneichuk N. P., Shevchuk I. A., Ul'yanov P. L.
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.
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.
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
Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications
Babenko V. F., Kofanov V. A., Korneichuk N. P., Pichugov S. A.
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.
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.
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.
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.
Anatolii Mikhailovich Samoilenko (on his 60th birthday)
Berezansky Yu. M., Boichuk О. A., Korneichuk N. P., Korolyuk V. S., Koshlyakov V. N., Kulik V. L., Luchka A. Y., Mitropolskiy Yu. A., Pelyukh G. P., Perestyuk N. A., Skorokhod A. V., Skrypnik I. V., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4
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.
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 $H^α,\; 0 < α < 1$).
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.
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.
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.
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.
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.
Some problems of coding and reconstructing functions
Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 514-524
A derivation of exact estimates for the derivative of the spline-interpolation error
Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 206-210
Behavior of the derivatives of the error of a spline interpolation
Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 67–72
Approximation theory and optimization problems
Ukr. Mat. Zh. - 1990. - 42, № 5. - pp. 579–593
Optimal coding of vector-functions
Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 737-743
Optimal coding of elements of a metric space
Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 168–173
Approximation of differential functions and their derivatives by parabolic splines
Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 702-710
Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 804–805
Approximations by local splines of minimal defect
Ukr. Mat. Zh. - 1982. - 34, № 5. - pp. 617—621
Error bound of spline interpolation in an integral metric
Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 391–394
Inequalities for best spline approximation of periodic differentiable functions
Ukr. Mat. Zh. - 1979. - 31, № 4. - pp. 380–388
On the approximation of continuous functions by algebraic polynomials
Korneichuk N. P., Polovina A. I.
Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 326—339
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