Oleksiy Dovgoshey

National Academy of Sciences of Ukraine | ISP · Institute of Mathematics

In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the ob...

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojic has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under con...

Let X, Y be sets and let \(\Phi, \Psi\) be mappings with the domains X² and Y² respectively. We say that \(\Phi\) is combinatorially similar to \(\Psi\) if there are bijections \(f \colon \Phi(X^2) \to \Psi(Y^{2})\) and \(g \colon Y \to X\) such that \(\Psi(x, y) = f(\Phi(g(x), g(y)))\) for all \(x, y \in Y\). It is shown that the semigroups of bin...

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y \to X$ such that $\Psi(x, y) = f(\Phi(g(x), g(y)))$ for all $x$, $y \in Y$. Conditions under which a given mappi...

Several new characterizations of pseudoultrametric spaces and ultrametric spaces are found. In particular, a dual form of Pongsriiam-Termwuttipong characterization of the ultrametric-preserving functions is described.

The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf B_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $(\mathbf B_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtaine...

Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$ be a scaling sequence of positive real numbers tending to infinity. We define the pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity as a metric space whose points are equivalence classes of sequences $(x_n)_{n\in\mathbb N}\subset X$ which tend to in...

A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| \geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2...

A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.

We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions and essentially almost subharmonic functions. It is shown that the sum of two quasinearly subharmonic functions may not be quasinearly subharmonic....

In this paper, we generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence these of sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded m...

Some extremal properties of finite ultrametric spaces and related properties of representing trees are described. We also describe conditions under which the isomorphism of representing trees is equivalent to the isometricity of corresponding finite ultrametric spaces.

Background. We investigate the relationship between the boundedness of Lebesgue constants for the Lagrange polynomial interpolation on a compact subset of \[\mathbb R\] and the existence of a Faber basis in the space of continuous functions on this compact set. Objective. The aim of the paper is to describe the conditions on the matrix of interp...

Let $B_{1}(\Omega, \mathbb R)$ be the first Baire class of real functions in the pluri-fine topology on an open set $\Omega \subseteq \mathbb C^{n}$ and let $H_{1}^{*}(\Omega, \mathbb R)$ be the first functional Lebesgue class of real functions in the same topology. We prove the equality $B_{1}(\Omega, \mathbb R)=H_{1}^{*}(\Omega, \mathbb R)$ and s...

For subsets of $\mathbb R^+ = [0,\infty)$ we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at $0$ sets is strongly porous if and only if these sets are coherently porous. This result leads...

The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y,$ but there are no proper subsets of $Y,$ which satisfy this property. We find sufficient conditions under which $\mathfrak{M}$ has a minimal $\mathfrak{M}$-universal metric space and...

For subsets of $\mathbb{R}^+$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function $\mu:\mathbb{N} \to \mathbb{R}^+$ provides an extension of the concept of porosity numbers on subsets of $\mathbb{N}$. The main results describe interconnecti...

We define and study, for subsets of [0,∞) several types of strong right upper porosity at the point 0. Some characterizations of these types of porosity are obtained, including a characterization in terms of a universal property and a characterization in terms of a structural property.

Let $(X,d)$ be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality $|Sp(X)|\leqslant |X|$ where $Sp(X)=\{d(x,y)\colon x,y \in X\}$. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the numbe...

Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is a metric space consisting of some equivalence classes of convergent to $p$ sequences $(x_n), x_n \in X,$ whose degree of convergence is comparable with a given scaling sequence $(r_n), r_n\downarrow 0.$ A scaling sequence $(r_n)$ is normal if this sequence is eventually de...

Let $SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb R^{+}$ and let $\hat I(SP)$ be the intersection of maximal ideals $I \subseteq SP$. Some characteristic properties of sets $E\in\hat I(SP)$ are obtained. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $\mathbb R^{+}$ is a prop...

Let SP be the set of upper strongly porous at 0 subsets of ℝ+ and let Î(SP) be the intersection of maximal ideals I ⊆ SP. Some characteristic properties of sets E ∈ Î(SP) are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal g...

Let (X,d X ) and (Y,d Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F: X→Y is a weak similarity if it is surjective and there exists a strictly increasing f: D(Y)→D(X) such that d X =f∘d Y ∘(F⊗F). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F: X...

Let $X$ be a nonempty set and $\mathcal{F}(X)$ be the set of nonempty finite subsets of $X$. The paper deals with the extended metrics $\tau:\mathcal{F}(X)\to\mathbb{R}$ recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets $A$ with $|A|\leqslant n$ make possible to consider "distance functions" with $n$ vari...

A set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an extension to an isotone mapping from X to Y are found.

The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the catheterization of nonnegative, quasinearly subharmonic functions by mean value ineq...

Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence is comparable with a given scaling sequence (r_n), r_n\downarrow 0. We say that (r_n) is normal if there is (x_n) such that |d(x_n,p)-r_n|=o(r_n) for...

We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric cpaces with Busemann convex pretangent spaces is also described.

It was proved by Gomori and Hu in 1961 that for every finite nonempty ultrametric space $(X,d)$ the following inequality $|\Sp(X)|\leqslant |X|-1$ holds with $\Sp(X)=\{d(x,y):x,y \in X, x\neq y\}$. We characterize the spaces $X$, for which the equality in this inequality is attained by the structural properties of some graphs and show that the set...

Let (X,d,p) be a metric space with a metric d and a marked point p. We define the set of w-strongly porous at 0 subsets of [0,\infty) and prove that the distance set {d(x,p): x\in X} is w-strongly porous at 0 if and only if every pretangent space to X at p is bounded.

A criterion for the ultrametricity of pretangent spaces to general metric spaces is obtained.

We define and study the completely strongly porous at 0 subsets of R^{+}. Several characterizations of these subsets are obtained, among them the description via an universal property and structural one.

We study the statistical convergence of metric valued sequences and of their subsequences. The interplay between the statistical and usual convergences in metric spaces is also studied.

Let $A\subseteq\mathbb C$ be a starlike set with a center $a$. We prove that every tangent space to $A$ at the point $a$ is isometric to the smallest closed cone, with the vertex $a$, which includes $A$. A partial converse to this result is obtained. The tangent space to convex sets is also discussed.

Let ℝ+ = [0,∞) and let A ⊆ ℝ+n. We have found the necessary and sufficient conditions under which a function Φ: A → ℝ+ has an isotone subadditive continuation on ℝ+n. It allows us to describe the metrics, defined on the Cartesian product X 1×...×X n of given metric spaces $$\left( {X_1 ,d_{X_1 } } \right), \ldots ,\left( {X_n ,d_{X_n } } \right)$$,...

Let (G,w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G)-->R^+ can be extended to a pseudoultrametric on V(G) are found. A criterion of the uniqueness of this extension is also obtained. It is proved that G is complete k-partite with k >= 2 if and only if, for every pseudoultrametrizable weight w, there e...

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function f:F(X)-->R there is an ultrametric on X such that f(A)=diam A for every A\in F(X). For finite nondegenerate ultrametric spaces (X,d) it is shown that X together with the subset of diametrical pairs of poi...

We find necessary and sufficient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a ∈ X. Applying this general result we show that each logarithmic spiral has a unique pretangent space at the asymptotic point. Unbounded multiplicative subgroups of C * = C \ {0} having unique pretangent spaces at zero ar...

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

We find a set of necessary and sufficient conditions under which the weight $w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the gr...

We introduce a tangent space at an arbitrary point of a general metric space. It is proved that all tangent spaces are complete. The conditions under which these spaces have a finite cardinality are found. AMS 2010 Subject Classification: 54E35.

The paper deals with pretangent spaces to general metric spaces. An ultrametricity criterion for pretangent spaces is found and it is closely related to the metric betweenness in the pretangent spaces. Key wordsmetric spaces-pretangent spaces-ultrametric spaces-metric betweenness

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under con...

The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasi-nearly subharmonic functions generalize subharmonic functions. We give generalized mean value inequalities for quasinearly subharmonic functions, where, in addition to balls, we consider, as examples, certain John domains with Ahlfors regular bou...

We describe metric spaces with bounded pretangent spaces and characterize proper metric spaces with proper tangent spaces. We also present the necessary and sufficient conditions under which a tangent space is compact and build a compact ultrametric space X such that some pretangent space to X has the density c.

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under con...

We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into $$ \mathbb{R} $$ if every three-point subspace of X is isometrically imbedded into $$ \mathbb{R} $$. A series of corollaries of this theorem is obtained. We establish new criteria for...

We investigate a tangent space at a point of a general metric space and metric space valued derivatives. The conditions under which two different subspace of a metric space have isometric tangent spaces in a common point of these subspaces are completely determinated.

We find necessary and sufficient conditions under which an arbitrary metric space $X$ has a unique pretangent space at the marked point $a\in X$. Key words: Metric spaces; Tangent spaces to metric spaces; Uniqueness of tangent metric spaces; Tangent space to the Cantor set.

We describe some Cartesian products of metric spaces and find conditions under which products of ultrametric spaces are ultrametric.

We study the properties of real functions f for which the compositions f ◦ d is a metric for every metric space (X, d). The explicit form is found for the invertible elements of the semigroup F{\mathcal F} of all such functions. The increasing functions f Î F{f \in \mathcal F} are characterized by the subadditivity condition and a maximal inverse s...

We obtain some refinements and extensions of the Basic Covering Theorem in a metric space (X, ρ). The properties of the metric ρ are used to define an inclusion coefficient k in this theorem, and this is related to the supremum of numbers t such that ρ t is a metric in X. The inclusion coefficient k characterizes ultrametric spaces.

We introduce a tangent space at an arbitrary point of a general metric space. It is proved that all tangent spaces are complete. The conditions under which these spaces have a flnite cardinality are found.

We study rectifiable curves given by mutually singular coordinate functions in finite-dimensional normed spaces. We describe these curves in terms of the behaviour of approximative tangents and find a simple formula for their lengths. We deduce from these results new necessary and sufficient conditions for the mutual singularity of finitely many fu...

This is an attempt to give a systematic survey of properties of the famous Cantor ternary function.

Let f be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim(E) and ℋ s (E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim(f(E))=αdim(E) holds for each E⊆X. The problem is studied also for the Cantor ternary function...

GENERAL BIEBERBACH POLYNOMIALS For a finite Borel measure µ and a point λ ∈ C we define the analogue of the Bieberbach polynomial π n as a solution of a suitable extremal problem. We prove a L 2 (µ) convergence of the sequence {π n } to a functioñfunctioñ ϕ and find a characteristic extremal property of˜ϕof˜ of˜ϕ. The conditions for˜ϕfor˜ for˜ϕ = 0...

Let $f$ be a mapping from a metric space $X$ to a metric space $Y$ , and let $\alpha$ be a positive real number. Write $dim(E)$ and $\mathcal{H}^s(E)$ for the Hausdorff dimension and the $s$-dimensional Hausdorff measure of a set $E$. We give sufficient conditions that the equality $dim(f(E)) = \alpha dim(E)$ holds for each $E \subseteq X$. The pro...

Let μ be a compactly supported finite Borel measure in ℂ, and let Πn be the space of holomorphic polynomials of degree at most n furnished with the norm of L 2(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Πn their values at a point z ∈ ℂ. The main results demonstrate ho...

Let G be a simply connected, bounded domain on the plane with the boundary and let P() be a uniform closure of polynomials on . It is shown that the Rudin-Carleson Theorem about analytic extensions from zero measure boundary sets is valid for P() if and only if G is a Caratheodory do- main and G does not separate the plane. These conditions are als...

Suppose that G is a bounded simply connected domain on the plane with boundary Í\subseteq òln( \tfracdma dw ) dw = - ¥\int {\ln \left( {\tfrac{{d\mu _a }}{{d\omega }}} \right)} d\omega = - \infty equivalent to the completeness of the polynomials in Lt() or to the unboundedness of the calculating functional p p (z0), where p is a polynomial in Lt...

The Perel'man's result according to which the first modulus of continuity of any real-analytic function f is a function analytic in a certain neighborhood of the origin is generalized to the case of arbitrary moduli of continuity of higher order.

We obtain conditions under which the modulus of continuity of a piecewise analytic function given on a closed interval of the real axis is an analytic function in a neighborhood of zero.

We prove that a three-term recurrence relation for analytic polynomials orthogonal with respect to harmonic measure in a simply connected domain G exists if and only if G is an ellipse.

Let M be an arbitrary subset of the complex plane C, and let a function f; be defined on M. The connection between the holomorphy of f; and the degree of the best polynomial approximation of f; on compact subsets of M having a unique limit point is described.

For compact sets K ⊆ C of zero logarithmic capacity we obtain the analogue of the well-known Bernstein-Walsh theorem that describes the relation between the decreasing rate of the best uniform polynomial approximation to a function f on K and the possibility of its analytic extension into the "canonical" domain bounded by the level line of the Gree...

We describe the domain of analyticity of a continuous function f in terms of the sequence of the best polynomial approximations of f on a compact set K(K ⊂ ℂ) and the sequence of norms of Chebyshev polynomials for K.

For an arbitrary compact setK⊂ℂ, we relate the order and the type of an entire functionf to the sequenceE n (f,K) of best polynomial approximations to this function onK.

A space of holomorphic functions is considered. A topology in this space is “intermediate” between the topology of uniform convergence and the topology of uniform convergence on compact sets. The properties of systems of orthonormal polynomials are studied in Hilbert spaces with this topology.

The list of such spaces can be continued in different ways, for instance, we may put Q(G) = HP(G) N Lr(G) and Ilflle = ~1 IlfllHp + '~2 IIfIIL r, ~j > 0, j = 1, 2, and introduce weight functions in Examples 2 and 3 or consider other compactifications in Example 1. In this paper we give criteria for bounded rational and polynomial approximation in t...

The problem on the simultaneous approximation of functions from the Hardy spaces by continuous functions on the Riemann sphere for a system of disjoint simply connected domains is considered. A criterion for the solvability of this problem is established.

The approximation of functions from Hardy classes by bounded analytic functions is investigated. A theorem is proved, characterizing the sets of functions with equiabsolutely continuous integrals as limit points of the family of bounded subsets of the space H8.

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H1 by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire functio...

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H1 by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire functio...

We study the question of the connection between properties of the domain G and the density of polynomials in the Hardy classes HP(G) and also establish Jackson and Bernshtein type approximation theorems.

Let b be a complex number with |b|>1 and let D be a finite subset of the complex plane ℂ such that 0∈D and cardD≥2. A number z is representable by the system (D,b) if z=∑ j=-∞ M a j b j , where a j ∈D. We denote by F the set of numbers which are representable by (D,b) with M=-1. The set W consists of numbers that are (D,b) representable with a j =0...