https://vestnmath.dnu.dp.ua/index.php/rim/issue/feedResearches in Mathematics2021-01-11T15:23:28+00:00Editorial Mediatorvestmath@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 publishes, semiannually, research articles in basic areas of theoretical and applied mathematics, including, but not limited to: analysis, algebra, geometry, topology, probability theory and mathematical statistics, differential equations and mathematical physics.</p><p>The journal was published annually from 1963 to 1997 under the title “Researches on modern problems of summation and approximation of functions and their applications” and from 1998 to 2018 under the title “Dnipro University Mathematics Bulletin”.</p><p><em>Attention:</em> The template of the submission (.tex and .cls files) was updated on 03.05.2021. Please proceed to <a href="/index.php/rim/about/submissions#authorGuidelines">Author Guidelines</a> to download and use the latest version.</p></td></tr></tbody></table>https://vestnmath.dnu.dp.ua/index.php/rim/article/view/129Preamble2020-12-28T06:53:56+00:00Editors Editorsdnuvestmath@gmail.com2021-01-13T13:01:04+00:00Copyright (c) 2020 Editors Editorshttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/130Vitalii Pavlovych Motornyi (on occasion of the 80th birthday)2020-12-28T06:53:56+00:00V.F. Babenkodnuvestmath@gmail.comR.O. Bilichenkodnuvestmath@gmail.comS.V. Goncharovdnuvestmath@gmail.comO.V. Kovalenkodnuvestmath@gmail.comS.V. Konarevadnuvestmath@gmail.comV.O. Kofanovdnuvestmath@gmail.comT.Yu. Leskevychdnuvestmath@gmail.comN.V. Parfinovychdnuvestmath@gmail.comA.M. Paskodnuvestmath@gmail.comO.V. Polyakovdnuvestmath@gmail.comO.O. Rudenkodnuvestmath@gmail.comD.S. Skorokhodovdnuvestmath@gmail.comM.Ye. Tkachenkodnuvestmath@gmail.comV.M. Traktynskadnuvestmath@gmail.comM.B. Vakarchukdnuvestmath@gmail.com—2021-01-13T13:01:04+00:00Copyright (c) 2020 V.F. Babenko, R.O. Bilichenko, S.V. Goncharov, O.V. Kovalenko, S.V. Konareva, V.O. Kofanov, T.Yu. Leskevych, N.V. Parfinovych, A.M. Pasko, O.V. Polyakov, O.O. Rudenko, D.S. Skorokhodov, M.Ye. Tkachenko, V.M. Traktynska, M.B. Vakarchukhttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/131Kolmogorov inequalities for norms of Marchaud-type fractional derivatives of multivariate functions2021-01-11T15:23:28+00:00N.V. Parfinovychnat-vic-par@i.uaV.V. Pylypenkovpylypenko@mmf.dnulive.dp.uaWe obtain new sharp Kolmogorov type inequalities, estimating the norm of mixed Marchaud type derivative of multivariate function through the C-norm of function itself and its norms in Hölder spaces.2021-01-13T13:01:04+00:00Copyright (c) 2020 N.V. Parfinovych, V.V. Pylypenkohttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/132Recovery of continuous functions from their Fourier coefficients known with error2021-01-06T19:04:22+00:00K.V. Pozharskakate.shvai@gmail.comO.A. Pozharskyipozharskyio@gmail.comThe problem of optimal recovery is considered for functions from their Fourier coefficients known with error. In a more general statement,<br />this problem for the classes of smooth and atalytic functions defined on various compact manifolds can be found in the classical paper by<br />G.G. Magaril-Il'yaev, K.Y. Osipenko.<br /><br />Namely, the paper is devoted to the recovery of continuous real-valued functions $y$ of one variable from the classes $$$W^{\psi}_{p}$$$, $$$1 \leq p< \infty$$$,<br />that are defined in terms of generalized smoothness $$$\psi$$$ from their Fourier coefficients with respect to some complete orthonormal in the space $$$L_2$$$ system<br />$$$\Phi = \{ \varphi_k \}_{k=1}^{\infty}$$$ of continuous functions, that are blurred by noise.<br /><br />Assume that for function $$$y$$$ we know the values $$$y_k^{\delta}$$$ of their noisy Fourier coefficients, besides $$$y_k^{\delta} = y_k + \delta \xi_k$$$, $$$k = 1,2, \dots$$$,<br />where $$$y_k$$$ are the corresponding Fourier coefficients, $$$\delta \in (0,1)$$$, and $$$\xi = (\xi_k)_{k=1}^{\infty}$$$ is a noise.<br />Additionally let the functions from the system $$$\Phi$$$ be continuous and satisfy the condition $$$\| \varphi_k \|_{C}\leq C_1 k^{\beta}$$$, $$$k=1,2,\dots$$$,<br />where $$$C_1>0$$$, $$$\beta \geq 0$$$ are some constants, and $$$\| \cdot\|_{C}$$$ is the standart norm of the space $$$C$$$ of continuous on the segment $$$[0,1]$$$ functions.<br /><br />Under certain conditions on parameter $$$\psi$$$, we obtain order estimates of the approximation errors of functions from the classes<br />$$<br />W^{\psi}_{p} = \left\{ y \in L_2\colon \| y \|^p_{W^{\psi}_{p}} = \sum\limits_{k=1}^{\infty} \psi^p(k) |y_k|^p \leq 1 \right\}, \quad 1 \leq p< \infty,<br />$$<br />in metric of the space $$$C$$$ by the so-called<br />$$$\Lambda$$$-method of series summation that is defined by the number triangular matrix $$$\Lambda = \{ \lambda_k^n \}_{k=1}^n$$$, $$$n=n(\delta) \in \mathbb{N}$$$,<br />with some restrictions on its elements.<br /><br />Note, that we extend the known results [8, 7] to a more wide spectrum of the classes<br />of functions and for a more general restrictions on the noise level.<br />In our results a case is considered when the noise is stronger than those in the space $$$l_2$$$ of real number sequences, but not stochastic.2021-01-13T13:01:04+00:00Copyright (c) 2020 K.V. Pozharska, O.A. Pozharskyihttps://vestnmath.dnu.dp.ua/index.php/rim/article/view/133Properties of the second-kind Chebyshev polynomials of complex variable2021-01-06T19:05:46+00:00O.V. Veselovskaveselovskaov@gmail.comV.V. Dostoinav.dostoyna@gmail.comM.I. Klapchukm.klapchuk@gmail.comWe construct a system of functions biorthogonal with Chebyshev polynomials of the second kind on closed contours in the complex plane. Properties of these functions and sufficient conditions of expansion of analytic functions into series in Chebyshev polynomials of the second kind in complex domains are investigated. The examples of such expansions are given. In addition, combinatorial identities of self-interest are obtained.2021-01-13T13:01:04+00:00Copyright (c) 2020 O.V. Veselovska, V.V. Dostoina, M.I. Klapchuk