# Ruslan Khats’Drohobych Ivan Franko State Pedagogical University · Department of Mathematics and Economics

PhD

48
Publications
3,238
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183
Citations
Citations since 2017
19 Research Items
141 Citations
September 2004 - December 2010
Position
• Lecturer
Education
November 2002 - November 2005
Drohobych Ivan Franko State Pedagogical University
Field of study
• Mathematical Analysis
September 1996 - July 2002
Drohobych Ivan Franko State Pedagogical University
Field of study
• Mathematics and Physics

## Publications

Publications (48)
Article
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We present an example of linear differential operator in a Hilbert space, which has no eigenfunctions but has, in a certain sense, some generalized eigenfunctions. It is proved that this operator is formally adjoint to Bessel-type differential operators for which the systems of canonical eigenfunctions are over-complete. We also analyze the complet...
Article
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Let f be an entire function, f(0) = 1, F(z) = zf′ (z)/f(z), and Γm = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\bigcup_{j=1}^{m}\left\{z:\mathrm{arg}z={\psi }_{j}...
Article
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Let $f$ be an entire function, $f(0)=1$, $F(z)=zf^{\prime }(z)/f(z)$, and $\Gamma_m=\bigcup\limits_{j=1}^ m \{z: \arg z=\psi_{j}\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$. An entire function $f$ is called a function of improved regular growth if for some $\rho\in (0;+\infty)$ and $\rho_2\in (0;\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically...
Article
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In this paper, we study an integral representation of some class E 2,+ of even entire functions of exponential type σ ≤ 1. We also obtain an analog of the Paley-Wiener theorem related to the class E 2,+. In addition, we find necessary and sufficient conditions for the completeness of a system s k √ xs k J −3/2 (xs k) : k ∈ N in the space L 2 ((0; 1...
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We show an example of a linear differential operator in a Hilbert space that has no eigenfunctions but has some generalized eigenfunctions in a certain sense. We prove that this operator is formally adjoint to Bessel-type differential operators systems of canonical eigenfunctions of which are over-complete. We also investigate completeness of a sys...
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The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to...
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We establish asymptotic estimates for the logarithm and logarithmic derivative of a special canonical product with improved zero-distribution on a finite system of rays up to a limited value outside some exceptional sets. Besides, we investigate an asymptotic behavior of the derivative of a special canonical product at its zeros.In addition, we obt...
Article
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In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from thi...
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Let $f$ be an entire function with $f(0)=1$, $(\lambda_n)_{n\in\mathbb N}$ be the sequence of its zeros, $n(t)=\sum_{|\lambda_n|\le t}1$, $N(r)=\int_0^r t^{-1}n(t)\, dt$, $r>0$, $h(\varphi)$ be the indicator of $f$, and $F(z)=zf'(z)/f(z)$, $z=re^{i\varphi}$. An entire function $f$ is called a function of improved regular growth if for some $\rho\in... Article Full-text available Let$J_{\nu}$be the Bessel function of the first kind of index$\nu\ge 1/2$,$p\in\mathbb R$and$(\rho_k)_{k\in\mathbb N}$be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system$\big\{x^{-p-1}\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$in the weighted space$L^2((0;1);x^{2p} dx)$are f... Article Full-text available We establish the necessary and sufficient conditions for the completeness of the system$(x^{-p-1}\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\Bbb N)$in the space$L^2((0;1);x^{2p} dx)$, where$J_{\nu}$is the Bessel function of the first kind of index$\nu\ge 1/2$,$p\in\Bbb R$and$(\rho_k: k\in\Bbb N)$is an arbitrary sequence of distinct nonzero comple... Article Full-text available We find a criterion of unconditional basicity of the system$(\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N)$in the space$L^2(0;1)$where$J_{\nu}$is the Bessel function of the first kind of index$\nu\geq-1/2$and$(\rho_k:k\in\mathbb N)$is a sequence of distinct nonzero complex numbers. Article Full-text available We describe the asymptotic behavior of the logarithms of entire functions of improved regular growth with zeros on a finite system of rays in the metric of Lq[0, 2𝜋]. Article Full-text available Let f be an entire function of order ρ ∈ (0, +∞) with zeros on a finite system of rays {z : arg z = ψ j }, j ∈ {1,. .. , m}, 0 ≤ ψ 1 < ψ 2 <. .. < ψ m < 2π and h(ϕ) be its indicator. In [6], it has been proved that if f is of improved regular growth (an entire function f is called a function of improved regular growth if for some ρ ∈ (0, +∞) and ρ... Article Full-text available We describe asymptotic behavior of the logarithms of entire functions of improved regular growth with zeros on a finite system of rays in the metric of Lq [0, 2\pi ]. Описано асимптотичну поведiнку логарифмiв цiлих функцiй покращеного регулярного зростання з нулями на скiнченнiй системi променiв у Lq [0, 2\pi]-метрицi. Article Full-text available We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of rays in terms of Fourier coefficients of their logarithmic derivative. Preprint Full-text available We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of rays in terms of Fourier coefficients of their logarithmic derivative. Article Full-text available B. V. Vynnyts'kyi, R. V. Khats'. Complete biorthogonal systems of Bessel functions, Mat. Stud. 48 (2017), 150-155. Let ν ≥ −1/2 and (ρ k) k∈N be a sequence of nonzero complex numbers such that ρ 2 k ̸ = ρ 2 m for k ̸ = m. We prove that if the system { √ xρ k J ν (xρ k) : k ∈ N } of Bessel functions of the first kind of index ν ≥ −1/2 is exact (i.e.... Article Full-text available We find sufficient conditions for the basisness of the system ($$\sqrt {x\rho k} {J_v}\left( {x\rho k} \right):k \in N$$) in the space L2(0; 1) and establish a relationship between the approximation properties of this system and the properties of the system (τν+1/2E1/2(−τ2ρk2; μ): k ∈ N), where Jν is the Bessel function of the first kind of index ν... Article Full-text available Найдены достаточные условия базисности системы$(\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\Bbb N)$в пространстве$L^2(0;1)$и установлена взаимосвязь между аппроксимационными свойствами этой системы и свойствами системы$(\tau^{\nu+1/2}E_{1/2}(-\tau^2 \rho_k^2;\mu):k\in\Bbb N)$, где$J_{\nu}$-- функция Бесселя первого рода с индексом$\nu$и$E_{\rho}(...
Article
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We establish a criterion for the completeness and minimality of the system (x^{ −p−1} sqrt{ xρ_k} J_ν (xρ_k ) : k∈N) in the space L^2 ((0;1);x^2p dx) where J_ν is the Bessel function of the first kind of index ν>1/2, p ∈ R and (ρ_k : k ∈ N) is a sequence of distinct nonzero complex numbers.
Article
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We describe an asymptotic behavior of entire functions of improved regular growth with zeros on a finite system of rays in the metric of Lp[0;2pi].
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Введено поняття множини узагальнених власних значень та множини узагальнених власних векторів лінійного оператора. Наведені означення ми розглядаємо як метод пошуку біортогональної системи підсистеми власних векторів деяких лінійних операторів в гільбертовому просторі, системи класичних власних векторів яких є переповненими. Сказане ми демонструємо...
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Using the Fourier series method for entire functions, we investigate the asymptotic behavior of averaging of entire functions of improved regular growth.
Article
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We find the necessary and sufficient conditions for the completeness and minimality in the space $L^ 2 (01)$ of system $(\ sqrt {x\ rho_k} J_ {\ nu}(x\ rho_k): k\ in\ Bbb N)$ generated by Bessel function of the first kind of index $\ nu\ ge-1/2$. Moreover, we establish a criterion for the completeness and minimality of system \$(x^{-2}\ sqrt {x\...
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We establish a criterion for the improved regular growth of entire functions of positive order in terms of Fourier-Stieltjes coefficients of the sequence of their zeros that they are located on a finite system of rays.
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We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of half-lines in terms of their Fourier coefficients.
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Using a Fourier series method for entire functions, we find an asymptotics of averaging of entire functions of improved regular growth with zeros on a finite system of rays.
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Отримано деякі асимптотичні оцінки для спеціальної цілої функції за регулярного зростання послідовності її нулів.
Article
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Встановлено критерій покращеного регулярного зростання цілих функцій додатного порядку з нулями на скінченній системі променів в термінах їх коефіцієнтів Фур'є.
Article
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Let f be an entire function of order ρ∈(0;+∞) with the indicator h and assume that for some ρ 1 ∈(0;ρ) there exists an exceptional set U⊂C such that log|f(z)|=|z| ρ +o(|z| ρ 1 ), U∌z=re iφ →∞, and U can be covered by a system of pairwise disjoint disks U k ={z:|z-a k |<τ k },k∈ℕ, satisfying ∑ k∈ℕ τ k <+∞, ∑ k∈ℕ τ k |log(τ k )|<+∞. Then there exists...
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Отримано асимптотичні формули для логарифмічних похідних цілих функцій з покращеним розподілом нулів.
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We investigate an approximation properties of the systems of Bessel functions of index −3/2.
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We obtain new uniform asymptotic estimates for the logarithms of canonical products with zeros on a positive ray under the condition of regular growth of a zero-counting function.
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For a locally-integrable function ψ on the interval [1;+∞), we investigate the problem on interdependence between the conditions ψ(t)=Δt^ρ +o(t^ρ_1 ) (E∌t→+∞), ρ_1 ∈(0;ρ), and ∫1^r ψ(t)/t dt=Δρr^ρ +o(r^ρ_2 ) (r→+∞), ρ_2 ∈(0;ρ).
Thesis
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In the thesis we introduce a new concept of an entire function of improved regular growth and found a criterion for this regularity in the sense of zero distribution when the zeroes are located on a finite system of rays. We also obtain new asymptotic estimates for the Fourier coefficients of an entire function and for a canonical products. Besides...
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Using the Fourier series method for entire functions, we find asymptotics of averaging of entire functions of improved regular growth with zeros on a finite system of rays.
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We introduce a concept of an entire function of refined regular growth of nonintegral order. We found a criterion for this regularity in the sense of zero distribution when the zeroes are located on a finite system of rays.
Article
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Sufficient conditions under which for the Fourier coefficients S m (r;f) of an entire function f of the order ρ∈(0;+∞) with the indicator h the relation S m (r;f)=γ m r ρ +o(r ρ 3 ) (r→+∞), m∈ℤ, γ m =1 2pi∫ 0 2π h(φ)e -imφ dφ, holds true for some ρ 3 ∈(0;ρ), are found.
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For a canonical product L of entire order ρ∈(0;+∞) which equals the genus of the product we prove that if the counting function of zeros satisfies n(t)=Δt ρ +o(t ρ 1 ) (t→+∞), ρ 1 ∈(0,ρ), then ln|L(z)|=1 ρRez ρ ∑ λ n ≤r λ n -ρ -Δ|z| ρ 1 ρcosρφ+(φ-π)sinρφ+o(|z| ρ 2 ) (|z|→+∞), ρ 2 ∈(0;ρ).
Article
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For an entire function L of non-entire order ρ∈(0;+∞) we obtain a criterion on zeros under which ln|L(r k e iφ )|=πΔ sinπρr k ρ cosρ(φ-π)+o(r k ρ 2 ) (r k →+∞), 0<ρ 2 <ρ, holds on some circles {z:|z|=r k } uniformly in φ∈[0;2π).
Article
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For an entire function L of order ρ∈(0;1) conditions on zeros under which ln|L(re iφ )|=πΔ sinπρr ρ cosρ(φ-π)+O(1), z=re iφ →+∞, z∉E, are found.

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