Researches in Mathematics
https://vestnmath.dnu.dp.ua/index.php/rim
<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>Oles Honchar Dnipro National Universityen-USResearches in Mathematics2312-9557Authors who publish with this journal agree to the following terms:<ol><li>Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a <a href="http://creativecommons.org/licenses/by/3.0/" target="_new">Creative Commons Attribution License</a> that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.</li><li>Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.</li><li>Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See <a href="http://opcit.eprints.org/oacitation-biblio.html" target="_new">The Effect of Open Access</a>).</li></ol>Preamble
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/93
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2018-06-252018-06-2526112Grygorіj Oleksіjovych Gruzintsev: the man, the mathematician, the philosopher
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/94
The 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.M.B. VakarchukM.Ye. Tkachenko
Copyright (c) 2018 M.B. Vakarchuk, M.Ye. Tkachenko
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2018-06-252018-06-252613710.15421/241801The best polynomial approximations of some classes of analytic functions in the Hardy spaces
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/95
Problems 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.S.B. VakarchukV.I. ZabutnaM.B. Vakarchuk
Copyright (c) 2018 S.B. Vakarchuk, V.I. Zabutna, M.B. Vakarchuk
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2018-06-252018-06-2526181710.15421/241802On the convergence of multidimensional regular C-fractions with independent variables
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/96
In 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.R.I. Dmytryshyn
Copyright (c) 2018 R.I. Dmytryshyn
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2018-06-252018-06-25261182410.15421/241803Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/97
We 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 $$$\mu \{ \mathrm{supp}_{[a,b]} x^{(k)}_{\pm} \}$$$.V.V. KamenevaV.A. Kofanov
Copyright (c) 2018 V.V. Kameneva, V.A. Kofanov
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2018-06-252018-06-25261253610.15421/241804Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/98
We 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.O.V. Kozynenko
Copyright (c) 2018 O.V. Kozynenko
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2018-06-252018-06-25261374710.15421/241805Inequalities 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 domain
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/99
We 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.V.A. Kofanov
Copyright (c) 2018 V.A. Kofanov
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2018-06-252018-06-25261485510.15421/241806On mean approximation of function and its derivatives
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/100
Some properties of the functions being integrable on the segment were considered in this article. Estimates for approximation are obtained.V.P. Motornyi
Copyright (c) 2018 V.P. Motornyi
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2018-06-252018-06-25261566110.15421/241807Pointwise estimates of the best one-sided approximations of classes $$$W^r_{\infty}$$$ for $$$0 < r < 1$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/101
The asymptotic pointwise estimation of the best one-sided approximations to the classes $$$W^r_{\infty}$$$, $$$0 < r < 1$$$, has been established.A.M. Pas'koV.D. Stefura
Copyright (c) 2018 A.M. Pas'ko, V.D. Stefura
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2018-06-252018-06-25261626710.15421/241808On interpolation of operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$ in Lorentz spaces in borderline cases
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/102
The 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.B.I. PeleshenkoT.N. Semirenko
Copyright (c) 2018 B.I. Peleshenko, T.N. Semirenko
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2018-06-252018-06-25261687810.15421/241809On one property of zeros of polynomials of the least $$$(\alpha; \beta)$$$-deviation from zero in weighted space $$$L_p$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/103
We 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.O.V. Polyakov
Copyright (c) 2018 O.V. Polyakov
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2018-06-252018-06-25261798110.15421/241810The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/104
We 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 $$$N \rightarrow \infty$$$.D. Skorokhodov
Copyright (c) 2018 D. Skorokhodov
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2018-06-252018-06-25261828810.15421/241811The criterion of the best approximant for the multivariable functions in the space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$
https://vestnmath.dnu.dp.ua/index.php/rim/article/view/106
The 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.V.M. Traktyns'kaM.Ye. TkachenkoD.O. Osennikova
Copyright (c) 2018 V.M. Traktyns'ka, M.Ye. Tkachenko, D.O. Osennikova
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2018-06-252018-06-25261899410.15421/241812