In this paper we investigate the differential-operator equation $$ partial u (t, x) / partial t + varphi (i partial / partial x) u (t, x) = 0, quad (t, x) in (0, + infty) times mathbb {R} equiv Omega, $$ where the function $ varphi in C ^ {infty} (mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i partial / partial x $, in $ L_2 (mathbb {R}) $ it is established that the operator $ varphi (i partial / partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ partial u / partial t + sqrt {I- Delta} u = 0 $, $ Delta = D_x ^ 2 $, with the fractionation differentiation operator $ sqrt { I- Delta} = varphi (i partial / partial x) $, where $ varphi (sigma) = (1+ sigma ^ 2) ^ {1/2} $, $ sigma in mathbb {R} $ is attributed to the considered equation. Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, cdot) $ for $ t to + infty $ (solution stabilization) in spaces of type $ S '$.
本文研究了函数$ varphi in C ^ {infty} (mathbb {R}) $和满足一定条件的微分算子方程$$partial u (t, x) / partial t + varphi (i partial / partial x) u (t, x) = 0, quad (t, x) in (0, + infty) times mathbb {R} equiv Omega,$$。利用自伴随算子$ i partial / partial x $的谱函数的显式形式，在$ L_2 (mathbb {R}) $中建立了算子$ varphi (i partial / partial x) $可以理解为某一$ S $型空间中的伪微分算子。演化方程$ partial u / partial t + sqrt {I- Delta} u = 0 $, $ Delta = D_x ^ 2 $，带有分馏微分算子$ sqrt { I- Delta} = varphi (i partial / partial x) $，其中$ varphi (sigma) = (1+ sigma ^ 2) ^ {1/2} $, $ sigma in mathbb {R} $属于所考虑的方程。所考虑的方程是一个具有初始函数$ f $的非局部多点问题，该初始函数是类型为$ S $的空间或类型为$ S '$的空间的元素，该空间与类型为$ S $的空间拓扑共轭。建立了该问题的基本解的性质，证明了该问题在半空间$ t> 0 $中的正确可解性，得到了该问题的基本解与初始函数卷积的表示形式，得到了$ t to + infty $(解稳定)在$ S '$型空间中的解$ u (t, cdot) $的行为。
{"title":"THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS","authors":"R. Kolisnyk, V. Gorodetskyi, O. Martynyuk","doi":"10.31861/bmj2021.01.09","DOIUrl":"https://doi.org/10.31861/bmj2021.01.09","url":null,"abstract":"In this paper we investigate the differential-operator equation\u0000$$\u0000partial u (t, x) / partial t + varphi (i partial / partial x) u (t, x) = 0, quad (t, x) in (0, + infty) times mathbb {R} equiv Omega,\u0000$$\u0000where the function $ varphi in C ^ {infty} (mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i partial / partial x $, in $ L_2 (mathbb {R}) $ it is established that the operator $ varphi (i partial / partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ partial u / partial t + sqrt {I- Delta} u = 0 $, $ Delta = D_x ^ 2 $, with the fractionation differentiation operator $ sqrt { I- Delta} = varphi (i partial / partial x) $, where $ varphi (sigma) = (1+ sigma ^ 2) ^ {1/2} $, $ sigma in mathbb {R} $ is attributed to the considered equation.\u0000\u0000Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, cdot) $ for $ t to + infty $ (solution stabilization) in spaces of type $ S '$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125226556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some nonlinear Stokes system is considered. The initial-boundary value problem for the system is investigated and the existence and uniqueness of the weak solution for the problem is proved.
考虑了一类非线性Stokes系统。研究了该系统的初边值问题，证明了该问题弱解的存在唯一性。
{"title":"STOKES SYSTEM WITH VARIABLE EXPONENTS OF NONLINEARITY","authors":"O. Buhrii, M. Khoma","doi":"10.31861/bmj2022.02.03","DOIUrl":"https://doi.org/10.31861/bmj2022.02.03","url":null,"abstract":"Some nonlinear Stokes system is considered. The initial-boundary value problem for the system is investigated and the existence and uniqueness of the weak solution for the problem is proved.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125809973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a differential equation of the form $u'(t) + Au(t) = f(t), t in (0,infty)$, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup of linear operators in a Banach space $mathfrak{B}, f(t)$ is a $mathfrak{B}$-valued polynomial, the behavior in the preassigned points of solutions of the Cauchy problem $u(0) = u_{0} in mathfrak{B}$ depending on $f(t)$ is investigated.
对于形式为$u'(t) + Au(t) = f(t), t in (0,infty)$的微分方程，其中$A$是有界解析的无穷小生成器$C_{0}$ - Banach空间中的线性算子半群$mathfrak{B}, f(t)$是一个$mathfrak{B}$值多项式，研究了柯西问题$u(0) = u_{0} in mathfrak{B}$依赖于$f(t)$的解在预分配点上的行为。
{"title":"ON SOLUTIONS OF THE NONHOMOGENEOUS CAUCHY PROBLEM FOR PARABOLIC TYPE DIFFERENTIAL EQUATIONS IN A BANACH SPACE","authors":"V. Gorbachuk","doi":"10.31861/bmj2022.02.02","DOIUrl":"https://doi.org/10.31861/bmj2022.02.02","url":null,"abstract":"For a differential equation of the form $u'(t) + Au(t) = f(t), t in (0,infty)$, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup of linear operators in a Banach space $mathfrak{B}, f(t)$ is a $mathfrak{B}$-valued polynomial, the behavior in the preassigned points of solutions of the Cauchy problem $u(0) = u_{0} in mathfrak{B}$ depending on $f(t)$ is investigated.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121849705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The potential theory method was used to study the existence of a solution of a multi- point boundary value problem for a 2b-parabolic equation. Using the Green’s function of a homogeneous boundary value problem for a 2b-parabolic equation, the integral Fredholm equation of the second kind is placed in accordance with the multipoint boundary value problem. Taking into account the constraints on the coefficients of the nonlocal condition and using the sequential approximation method, an integrated image of the solution of the nonlocal problem at the initial moment of time and its estimation in the Holder spaces are found. Estimates of the solution of a nonlocal multipoint boundary value problem at fixed moments of time given in a nonlocal condition are found by means of estimates of the components of the Green’s function of the general boundary value problem for the 2b-parabolic equation. Taking into account the obtained estimates and constraints on coefficients in multipoint problem, estimates of the solution of the multipoint problem for the 2b-parabolic equations and its derivatives in Holder spaces are established. In addition, the uniqueness and integral image of the solution of the general multipoint problem for 2b-parabolic equations is justified. The obtained result is applied to the study of the optimal system control problem described by the general multipoint boundary value problem for 2b-parabolic equations. The case of simultaneous internal, initial and boundary value control of solutions to a multipoint parabolic boundary value problem is considered. The quality criterion is defined by the sum of volume and surface integrals. The necessary and sufficient conditions for the existence of an optimal solution of the system described by the general multipoint boundary value problem for 2b-parabolic equations with limited internal, initial and boundary value control are established.
{"title":"OPTIMAL CONTROL IN THE MULTIPOINT BOUNDARY VALUE PROBLEM FOR 2B-PARABOLIC EQUATIONS","authors":"I. Pukalskyi, I. Luste","doi":"10.31861/bmj2022.01.10","DOIUrl":"https://doi.org/10.31861/bmj2022.01.10","url":null,"abstract":"The potential theory method was used to study the existence of a solution of a multi-\u0000point boundary value problem for a 2b-parabolic equation. Using the Green’s function of a\u0000homogeneous boundary value problem for a 2b-parabolic equation, the integral Fredholm equation of the second kind is placed in accordance with the multipoint boundary value problem.\u0000Taking into account the constraints on the coefficients of the nonlocal condition and using the\u0000sequential approximation method, an integrated image of the solution of the nonlocal problem\u0000at the initial moment of time and its estimation in the Holder spaces are found. Estimates of\u0000the solution of a nonlocal multipoint boundary value problem at fixed moments of time given in\u0000a nonlocal condition are found by means of estimates of the components of the Green’s function of the general boundary value problem for the 2b-parabolic equation. Taking into account\u0000the obtained estimates and constraints on coefficients in multipoint problem, estimates of the\u0000solution of the multipoint problem for the 2b-parabolic equations and its derivatives in Holder\u0000spaces are established. In addition, the uniqueness and integral image of the solution of the\u0000general multipoint problem for 2b-parabolic equations is justified. The obtained result is applied to the study of the optimal system control problem described by the general multipoint\u0000boundary value problem for 2b-parabolic equations. The case of simultaneous internal, initial\u0000and boundary value control of solutions to a multipoint parabolic boundary value problem is\u0000considered. The quality criterion is defined by the sum of volume and surface integrals. The\u0000necessary and sufficient conditions for the existence of an optimal solution of the system described by the general multipoint boundary value problem for 2b-parabolic equations with limited\u0000internal, initial and boundary value control are established.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130760212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we show that a non-degenerate monodromic critical point of differential systems with the line at infinity and an affine real invariant straight line of total multiplicity four is a center type if and only if the first four Lyapunov quantities vanish.
{"title":"CENTER PROBLEM FOR CUBIC DIFFERENTIAL SYSTEMS WITH THE LINE AT INFINITY AND AN AFFINE REAL INVARIANT STRAIGHT LINE OF TOTAL MULTIPLICITY FOUR","authors":"A. Suba, O. Vacaras","doi":"10.31861/bmj2021.02.03","DOIUrl":"https://doi.org/10.31861/bmj2021.02.03","url":null,"abstract":"In this article, we show that a non-degenerate monodromic critical point of differential\u0000systems with the line at infinity and an affine real invariant straight line of total multiplicity\u0000four is a center type if and only if the first four Lyapunov quantities vanish.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132881326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We find conditions for a singular point O(0, 0) of a center or a focus type to be a center, in a cubic differential system with one irreducible invariant cubic. The presence of a center at O(0, 0) is proved by constructing integrating factors.
{"title":"CENTER CONDITIONS FOR A CUBIC DIFFERENTIAL SYSTEM HAVING AN INTEGRATING FACTOR","authors":"D. Cozma, A. Matei","doi":"10.31861/bmj2020.02.01","DOIUrl":"https://doi.org/10.31861/bmj2020.02.01","url":null,"abstract":"We find conditions for a singular point O(0, 0) of a center or a focus type to be a center,\u0000in a cubic differential system with one irreducible invariant cubic. The presence of a center at O(0, 0) is proved by constructing integrating factors.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133079416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Instability of unbounded solutions of evolution equations with operator coefficients commuting with rotation operators","authors":"V. Slyusarchuk","doi":"10.31861/bmj2019.01.099","DOIUrl":"https://doi.org/10.31861/bmj2019.01.099","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122035156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lyapunov’s second method is used to study the problem of stability of controlled stochastic dynamical systems of random structure with Markov and Poisson perturbations. Markov switches reflect random effects on the system at fixed points in time. Poisson perturbations describe random effects on the system at random times. In both cases there may be breaks in the phase trajectory of the first kind. The conditions for the coefficients of the system are written, which guarantee the existence and uniqueness of the solution of the stochastic system of a random structure, which is under the action of Markov switches and Poisson perturbations. The differences between these systems and systems that do not contain internal perturbations in the equation, which cause a change in the structure of the system, and external perturbations, which cause breaks in the phase trajectory at fixed points in time, are discussed. The upper bound of the solution for the norm is obtained. The definition of the discrete Lyapunov operator based on the system and the Lyapunov function for the above-mentioned systems is given. Sufficient conditions of asymptotic stochastic stability in general, stability in l.i.m. and asymptotic stability in the l.i.m. for controlled stochastic dynamic systems of random structure with Markov switches and Poisson perturbations are obtained. A model example that reflects the features of the stability of the solution of a system with perturbations is considered: the conditions of asymptotic stability in the root mean square as a whole are established; the conditions of exponential stability and exponential instability are discussed. For linear systems, the necessary and sufficient stability conditions are determined in the example, based on the generalized Lyapunov exponent.
{"title":"STABILITY OF CONTROLLED STOCHASTIC DYNAMIC SYSTEMS OF RANDOM STRUCTURE WITH MARKOV SWITCHES AND POISSON PERTURBATIONS","authors":"T. Lukashiv, I. Malyk","doi":"10.31861/bmj2022.01.08","DOIUrl":"https://doi.org/10.31861/bmj2022.01.08","url":null,"abstract":"Lyapunov’s second method is used to study the problem of stability of controlled\u0000stochastic dynamical systems of random structure with Markov and Poisson perturbations. Markov switches reflect random effects on the system at fixed points in time.\u0000Poisson perturbations describe random effects on the system at random times. In both\u0000cases there may be breaks in the phase trajectory of the first kind.\u0000The conditions for the coefficients of the system are written, which guarantee the\u0000existence and uniqueness of the solution of the stochastic system of a random structure, which is under the action of Markov switches and Poisson perturbations. The differences between these systems and systems that do not contain internal perturbations in the equation, which cause a change in the structure of the system, and external perturbations, which cause breaks in the phase trajectory at fixed points in time, are discussed. The upper bound of the solution for the norm is obtained. The definition of the discrete Lyapunov operator based on the system and the Lyapunov function for the above-mentioned systems is given.\u0000Sufficient conditions of asymptotic stochastic stability in general, stability in l.i.m.\u0000and asymptotic stability in the l.i.m. for controlled stochastic dynamic systems of random structure with Markov switches and Poisson perturbations are obtained.\u0000A model example that reflects the features of the stability of the solution of a system\u0000with perturbations is considered: the conditions of asymptotic stability in the root mean\u0000square as a whole are established; the conditions of exponential stability and exponential instability are discussed. For linear systems, the necessary and sufficient stability conditions are determined in the example, based on the generalized Lyapunov exponent.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126711142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Y. Goncharenko, M. Pratsiovytyi, S. Dmytrenko, I. Lysenko, S. Ratushniak
We consider one generalization of functions, which are called as «binary self-similar functi- ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, differential and fractal properties are studied for the functions of this class.
{"title":"ABOUT ONE CLASS OF FUNCTIONS WITH FRACTAL PROPERTIES","authors":"Y. Goncharenko, M. Pratsiovytyi, S. Dmytrenko, I. Lysenko, S. Ratushniak","doi":"10.31861/bmj2021.01.23","DOIUrl":"https://doi.org/10.31861/bmj2021.01.23","url":null,"abstract":"We consider one generalization of functions, which are called as «binary self-similar functi-\u0000ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, differential and fractal properties are studied for the functions of this class.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124768005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.
{"title":"APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS","authors":"JohnRichens, 郭利劭","doi":"10.31861/BMJ2020.02.10","DOIUrl":"https://doi.org/10.31861/BMJ2020.02.10","url":null,"abstract":"The paper is devoted to the approximation by arithmetic means of Fourier sums of classes\u0000of periodic functions of high smoothness. The simplest example of a linear approximation\u0000of continuous periodic functions of a real variable is the approximation by partial sums of the\u0000Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class\u0000of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to\u0000construct trigonometrical polynomials that would be uniformly convergent for each continuous\u0000function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely\u0000studied. One of the most important direction in this field is the study of the asymptotic behavior\u0000of upper bounds of deviations of linear means of Fourier sums on different classes of periodic\u0000functions. Methods of investigation of integral representations of deviations of trigonometric\u0000polynomials generated by linear methods of summation of Fourier series, were originated and\u0000developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and\u0000others.\u0000The aim of the work systematizes known results related to the approximation of classes\u0000of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for\u0000particular cases. In the paper is studied the approximative properties of repeated Fejer sums on\u0000the classes of periodic analytic functions of real variable. Under certain conditions, we obtained\u0000asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson\u0000integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky\u0000problem without any additional conditions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129637622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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