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Oleg Kovalenko

Publications and source records attributed to Oleg Kovalenko.

At least 19 recordsLinked to original sources
arXiv2026

A note on the Kolmogorov-type inequalities for more than three norms

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm $\|f^{(k)}\|$ of the $k$-th derivative of a function $f\colon \mathbb{R}\to\mathbb{R}$ by the values of the uniform norm of $f$ and uniform norms of several its higher derivatives ($\|f^{(r)}\|$ and $\|f^{(r-1)}\|$, or $\|f^{(r)}\|$ and $\|f^{(r-2)}\|$, or $\|f^{(r)}\|$, $\|f^{(r-1)}\|$ and $\|f^{(r-2)}\|$) using standard techniques can be obtained from the known solutions to the Kolmogorov problem about existence of a function with given norms of its derivatives.

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arXiv2024

Ostrowski-type inequalities in abstract distance spaces

For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If $(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance space, and $H(T,X)$ is a class of Lipschitz mappings $f\colon T\to X$, for a broad family of mappings $\Lambda\colon H (T,X)\to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(\Lambda f(\cdot),\Lambda f(t))$ in terms of the function $h_T(\cdot, t)$. We also show that many known estimates of such kind are contained in our general result.

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arXiv2023

On optimization of cubature formulae for Sobolev classes of functions defined on star domains

We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm of their distributional gradient; thus we generalize the known solution of this optimization problem in the case, when the domain of definition of the functions is convex.

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arXiv2023

On a general approach to some problems of approximation of operators

In the article we propose a general scheme for solutions of some approximation problems under a rather general setting. We illustrate the application of the proposed scheme by a series of examples, in particular we show that many results in the area of Ostrowski type inequalities can be obtained by standard arguments.

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arXiv2023

On discretization of some extremal problems

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class of functions; in the second one we find extremal values for the expectation of a random process with monotone trajectories at a random time. In both cases we reduce the continuous problems to their discrete counterparts. The obtained discrete problems are on the one hand interesting on their own, and on the other hand give a natural explanation of the structure of the extremal functions for the continuous problems.

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arXiv2023

Nagy type inequalities in metric measure spaces and some applications

We obtain a sharp Nagy type inequality in a metric space $(X,\rho)$ with measure $\mu$ that estimates the uniform norm of a function using its $\|\cdot\|_{H^\omega}$ -- norm determined by a modulus of continuity $\omega$, and a seminorm that is defined on a space of locally integrable functions. We consider charges $\nu$ that are defined on the set of $\mu$-measurable subsets of $X$ and are absolutely continuous with respect to $\mu$. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a $\|\cdot\|_{H^\omega}$-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case $X=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\|\cdot\|_{H^\omega}$-norm of its mixed derivative.

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arXiv2023

On Landau -- Kolmogorov type inequalities for charges and their applications

In this article we prove sharp Landau--Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $\mathbb{R}^d$, $d\geq 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon--Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$ belongs to the Sobolev space $W^{1,\infty}$.

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arXiv2022

Fixed sets and fixed points in $\Lim$--spaces]{Fixed sets and fixed points for mappings in generalized $\Lim$--spaces of Fr\'echet

In this article we discuss a possibility to implement a well-known scheme of proof for contraction mapping theorems in a situation, when convergence, families of Cauchy sequences, and contractiveness of mappings are defined axiomatically. We also consider ways to specify families of Cauchy sequences and contractiveness conditions using distance-like functions with values in some partially ordered set and establish fixed set and point theorems for generalized contractions of the \'Ciri\'c and Caristi types.

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arXiv2021

On approximation of hypersingular integral operators by bounded ones

We solve the Stechkin problem about approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular, we obtain sharp Landau-Kolmogorov type inequalities in both additive and multiplicative forms for hypersingular integral operators and prove a sharp Ostrowski type inequality for multivatiate Sobolev classes. We also give some applications of the obtained results, in particular study the modulus of continuity of the hypersingular integral operators, and solve the problem of optimal recovery of the value of a hypersingular integral operator based on the argument known with an error.

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arXiv2021

Fixed Points Theorems in Hausdorff M-distance Spaces

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including metric and uniform spaces. On the other hand, compared to the so-called cone metric spaces and $K$-metric spaces, we do not require that the distance function range has a linear structure. We also consider several applications of the obtained fixed point theorems. In particular, we consider the questions of the existence of solutions of the Fredholm integral equation in $L$-spaces.

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arXiv2021

Fixed Point Theorems in M-distance Spaces

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including metric and uniform spaces. On the other hand, compared to the so-called cone metric spaces and $K$-metric spaces, we do not require that the distance function range has a linear structure. We also consider several applications of the obtained fixed point theorems. In particular, we consider the questions of the existence of solutions of the Fredholm integral equation in $L$-spaces.

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arXiv2020

Korneichuk-Stechkin Lemma, Ostrowski and Landau inequalities, and optimal recovery problems for $L$-space Valued Functions

We prove an analogue of the Korneichuk--Stechkin lemma for functions with values in $L$-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery, on the classes of $L$-space valued functions with given majorant of modulus of continuity. The recovery is done based on $n$ mean values of the functions over some intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on $n$ values of the function. We also obtain some sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of $L$-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions as well as for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions.

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arXiv2020

On optimal recovery of integrals of random processes

In this paper we prove a sharp Ostrowski type inequality for random processes of certain classes. This inequality is later applied to a solution of the optimal recovery of the integral $\int_0^1ξ_tdt$, using the random variables $ξ_{τ_1},\dots, ξ_{τ_n}$ as an information set, where $τ_1,\dots, τ_n$ are random variables. We also consider the problem of the information set optimization.

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arXiv2018

On maximally oscillating perfect splines and some of their extremal properties

In this paper we study analogues of the perfect splines for weighted Sobolev classes of functions defined on the half-line. Maximally oscillating splines play important role in the solution of certain extremal problems. In particular, using these splines, we characterize the modulus of continuity of the differential operator.

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arXiv2014

Kolmogorov problem on the class of multiply monotone functions

Necessary and sufficient conditions for positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}$, $0 = k_1 < k_2<k_2\leq r-2$, $k_4=r$, to guarantee the existence of an $r-1$-monotone function defined on the negative half-line and such that $\|x^{(k_i)}\| = M_{k_i}$, $i=1,2,3,4$ were found.

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