https://vestnmath.dnu.dp.ua/index.php/rim/issue/feedResearches in Mathematics2018-07-20T17:50:29+00:00Rudenko O.O.vestmath@mmf.dnulive.dp.uaOpen Journal Systems<table><tbody><tr><td><img src="/files/pics/gen_issue_cover.png" alt="" width="100px" height="150px" /></td><td style="padding-left: 16px; vertical-align: top;"><p>The journal presents results of researches concerning algebra, approximation theory for the functions of real variable, the equations of mathematical physics, and their application to problem solution.</p><p>The journal is intended for research officers, post-graduate and undergraduate students.</p></td></tr></tbody></table>https://vestnmath.dnu.dp.ua/index.php/rim/article/view/93Preamble2018-07-20T17:47:52+00:00Editors Editorsdnuvestmath@gmail.com2018-06-25T15:57:54+00:00Copyright (c) 2018 Editors Editorshttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/94Grygorіj Oleksіjovych Gruzintsev: the man, the mathematician, the philosopher2018-07-20T17:47:52+00:00M.B. Vakarchukmihailvakarchuk@gmail.comM.Ye. Tkachenkomtkachenko2009@ukr.netThe article describes the main milestones of the life of the professor of mathematics G. О. Gruzintsev and some aspects of his research work in mathematics and in the theory of science. The list of the main works of G. О. Gruzintsev is given.2018-06-25T15:57:55+00:00Copyright (c) 2018 M.B. Vakarchuk, M.Ye. Tkachenkohttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/95The best polynomial approximations of some classes of analytic functions in the Hardy spaces2018-07-20T17:47:52+00:00S.B. Vakarchuksbvakarchuk@gmail.comV.I. Zabutnavizabutna@gmail.comM.B. Vakarchukmihailvakarchuk@gmail.comProblems of the best polynomial approximation of classes of analytic functions $$$H^m_{p,R}$$$, $$$m\in \mathbb{Z}_+$$$, $$$R \geqslant 1$$$, $$$1 \leqslant p \leqslant \infty$$$, have been investigated in the Hardy spaces $$$H_p$$$. The best linear methods of approximation were constructed on the indicated classes.2018-06-25T15:57:55+00:00Copyright (c) 2018 S.B. Vakarchuk, V.I. Zabutna, M.B. Vakarchukhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/96On the convergence of multidimensional regular C-fractions with independent variables2018-07-20T17:47:52+00:00R.I. Dmytryshyndmytryshynr@hotmail.comIn this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.2018-06-25T15:57:55+00:00Copyright (c) 2018 R.I. Dmytryshynhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/97Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain2018-07-20T17:47:52+00:00V.V. Kamenevavlada-kameneva@i.uaV.A. Kofanovvladimir.kofanov@gmail.comWe solve the extremal problem $$$\| x^{(k)}_{\pm} \|_{L_p[a,b]} \rightarrow \sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$x\in W^r_{\infty} (\mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$\mu \{ \mathrm{supp}_{[a,b]} x^{(k)}_{\pm} \}$$$.2018-06-25T15:57:56+00:00Copyright (c) 2018 V.V. Kameneva, V.A. Kofanovhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/98Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions2018-07-20T17:47:52+00:00O.V. Kozynenkokozinenkoalex@gmail.comWe consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.2018-06-25T15:57:56+00:00Copyright (c) 2018 O.V. Kozynenkohttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/99Inequalities of various metrics for the norms $$$\|x\|_{p,\delta} = \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a,b\in \mathbb{R}, b-a\leqslant \delta \bigr\}$$$ of differentiable functions on the real domain2018-07-20T17:47:52+00:00V.A. Kofanovvladimir.kofanov@gmail.comWe prove sharp inequalities of various metrics for the norms $$$\| x \|_{p, \delta}$$$ of differentiable functions defined on the real line, trigonometric polynomials and periodic splines.2018-06-25T15:57:56+00:00Copyright (c) 2018 V.A. Kofanovhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/100On mean approximation of function and its derivatives2018-07-20T17:47:52+00:00V.P. Motornyimotornyivp@yandex.ruSome properties of the functions being integrable on the segment were considered in this article. Estimates for approximation are obtained.2018-06-25T15:57:57+00:00Copyright (c) 2018 V.P. Motornyihttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/101Pointwise estimates of the best one-sided approximations of classes $$$W^r_{\infty}$$$ for $$$0 < r < 1$$$2018-07-20T17:47:52+00:00A.M. Pas'kopasko08@meta.uaV.D. Stefuradnuvestmath@gmail.comThe asymptotic pointwise estimation of the best one-sided approximations to the classes $$$W^r_{\infty}$$$, $$$0 < r < 1$$$, has been established.2018-06-25T15:57:57+00:00Copyright (c) 2018 A.M. Pas'ko, V.D. Stefurahttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/102On interpolation of operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$ in Lorentz spaces in borderline cases2018-07-20T17:47:52+00:00B.I. Peleshenkodsaupelesh@ukr.netT.N. Semirenkotsemyrenko@ukr.netThe quaslinear operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$, analogs of the Calderon, Bennett operators in the case of concave and convex functions $$$\phi_0(t)$$$, $$$\psi_0(t)$$$, $$$\phi_1(t)$$$, $$$\psi_1(t)$$$ are considered. The theorems of interpolation of these operators from the Lorentz space $$$\Lambda_{\psi, b}(\mathbb{R}^n)$$$ into the space $$$\Lambda_{\psi, a}(\mathbb{R}^n)$$$ in cases when $$$0 < b \leqslant a \leqslant 1$$$ and relation of function $$$\phi^{\frac{1}{b}}(t)$$$ to one of functions $$$\phi_1(t)$$$, $$$\phi_2(t)$$$ is slowly varying function are proved.2018-06-25T15:57:57+00:00Copyright (c) 2018 B.I. Peleshenko, T.N. Semirenkohttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/103On one property of zeros of polynomials of the least $$$(\alpha; \beta)$$$-deviation from zero in weighted space $$$L_p$$$2018-07-20T17:50:29+00:00O.V. Polyakovov_polyakov@mail.ruWe prove the property of monotonity of zeros of polynomials of the least $$$(\alpha; \beta)$$$-deviation from zero in the space with integral metrics with weight.2018-06-25T15:57:58+00:00Copyright (c) 2018 O.V. Polyakovhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/104The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions2018-07-20T17:47:53+00:00D. Skorokhodovdmitriy.skorokhodov@gmail.comWe show that the error of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions comprising $$$N$$$ elements has the order $$$N^{-2}$$$ as $$$N \rightarrow \infty$$$.2018-06-25T15:57:58+00:00Copyright (c) 2018 D. Skorokhodovhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/106The criterion of the best approximant for the multivariable functions in the space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$2018-07-20T17:47:53+00:00V.M. Traktyns'katraktynskaviktoriia@gmail.comM.Ye. Tkachenkomtkachenko2009@ukr.netD.O. Osennikovadasha.sasha.elena@gmail.comThe questions of the characterization of the best approximant for the multivariable functions in the space with mixed integral metric were considered in this article. The criterion of the best approximant in space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$ is obtained.2018-06-25T15:57:58+00:00Copyright (c) 2018 V.M. Traktyns'ka, M.Ye. Tkachenko, D.O. Osennikova