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.
{"title":"On the convergence of multidimensional regular C-fractions with independent variables","authors":"R. Dmytryshyn","doi":"10.15421/241803","DOIUrl":"https://doi.org/10.15421/241803","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122326306","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 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.
{"title":"On interpolation of operators of weak type $$$(phi_0, psi_0, phi_1, psi_1)$$$ in Lorentz spaces in borderline cases","authors":"B. I. Peleshenko, T. N. Semirenko","doi":"10.15421/241809","DOIUrl":"https://doi.org/10.15421/241809","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132242489","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 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.
我们证明了实线、三角多项式和周期样条上定义的可微函数的范数$$$ x _{p， delta}$$$ $的各种度量的尖锐不等式。
{"title":"Inequalities of various metrics for the norms $$$|x|_{p,delta} = sup bigl{ | x |_{L_p[a,b]} colon a,bin mathbb{R}, b-aleqslant delta bigr}$$$ of differentiable functions on the real domain","authors":"V. Kofanov","doi":"10.15421/241806","DOIUrl":"https://doi.org/10.15421/241806","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130560347","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}
Problems of the best polynomial approximation of classes of analytic functions $$$H^m_{p,R}$$$, $$$min mathbb{Z}_+$$$, $$$R geqslant 1$$$, $$$1 leqslant p leqslant infty$$$, have been investigated in the Hardy spaces $$$H_p$$$. The best linear methods of approximation were constructed on the indicated classes.
{"title":"The best polynomial approximations of some classes of analytic functions in the Hardy spaces","authors":"S. Vakarchuk, V. I. Zabutna, M. Vakarchuk","doi":"10.15421/241802","DOIUrl":"https://doi.org/10.15421/241802","url":null,"abstract":"Problems of the best polynomial approximation of classes of analytic functions $$$H^m_{p,R}$$$, $$$min mathbb{Z}_+$$$, $$$R geqslant 1$$$, $$$1 leqslant p leqslant infty$$$, have been investigated in the Hardy spaces $$$H_p$$$. The best linear methods of approximation were constructed on the indicated classes.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131535803","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 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.
证明了最小$$$( α;beta)$$$-在空间中与零的偏差，带有权重的积分度量。
{"title":"On one property of zeros of polynomials of the least $$$(alpha; beta)$$$-deviation from zero in weighted space $$$L_p$$$","authors":"O. V. Polyakov","doi":"10.15421/241810","DOIUrl":"https://doi.org/10.15421/241810","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"352 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126688057","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 asymptotic pointwise estimation of the best one-sided approximations to the classes $$$W^r_{infty}$$$, $$$0 < r < 1$$$, has been established.
建立了类$$$W^r_{infty}$$$， $$$0 < r < 1$$$ $的最佳单侧逼近的渐近点估计。
{"title":"Pointwise estimates of the best one-sided approximations of classes $$$W^r_{infty}$$$ for $$$0 < r < 1$$$","authors":"A. M. Pas'ko, V. D. Stefura","doi":"10.15421/241808","DOIUrl":"https://doi.org/10.15421/241808","url":null,"abstract":"The asymptotic pointwise estimation of the best one-sided approximations to the classes $$$W^r_{infty}$$$, $$$0 < r < 1$$$, has been established.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128949370","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 properties of the functions being integrable on the segment were considered in this article. Estimates for approximation are obtained.
本文研究了函数在段上可积的一些性质。得到了近似的估计。
{"title":"On mean approximation of function and its derivatives","authors":"V. P. Motornyi","doi":"10.15421/241807","DOIUrl":"https://doi.org/10.15421/241807","url":null,"abstract":"Some properties of the functions being integrable on the segment were considered in this article. Estimates for approximation are obtained.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"135 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116547963","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 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 $$$N rightarrow infty$$$.
{"title":"The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions","authors":"D. Skorokhodov","doi":"10.15421/241811","DOIUrl":"https://doi.org/10.15421/241811","url":null,"abstract":"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 $$$N rightarrow infty$$$.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121355142","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 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.
{"title":"Grygorіj Oleksіjovych Gruzintsev: the man, the mathematician, the philosopher","authors":"M. Vakarchuk, M. Tkachenko","doi":"10.15421/241801","DOIUrl":"https://doi.org/10.15421/241801","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128586994","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 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.
{"title":"The criterion of the best approximant for the multivariable functions in the space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$","authors":"V. M. Traktyns'ka, M. Tkachenko, D. O. Osennikova","doi":"10.15421/241812","DOIUrl":"https://doi.org/10.15421/241812","url":null,"abstract":"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.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"6 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113962084","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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