We have found exact values of deviation of even Hermitian splines on some classes of functions and pointed out the best choice of nodes at approximation of concrete functions by these splines.
我们找到了偶厄密样条在某些函数上的精确偏差值，并指出了用这些样条逼近具体函数时节点的最佳选择。
{"title":"On the best choice of nodes at interpolation of functions by even Hermitian splines","authors":"A. D. Malysheva","doi":"10.15421/247707","DOIUrl":"https://doi.org/10.15421/247707","url":null,"abstract":"We have found exact values of deviation of even Hermitian splines on some classes of functions and pointed out the best choice of nodes at approximation of concrete functions by these splines.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89828908","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 establish conditions of $|gamma|_p$- and $[gamma]_p$-summability in degree $p geqslant 1$ of series, associated with Fourier series, at the point where $gamma = | gamma_{nk} |$ is the matrix of transformation of series to sequence.
{"title":"Absolute and strong summability in degree $p geqslant 1$ of series, associated with Fourier series, by matrix methods","authors":"N. Polovina","doi":"10.15421/247718","DOIUrl":"https://doi.org/10.15421/247718","url":null,"abstract":"We establish conditions of $|gamma|_p$- and $[gamma]_p$-summability in degree $p geqslant 1$ of series, associated with Fourier series, at the point where $gamma = | gamma_{nk} |$ is the matrix of transformation of series to sequence.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75527486","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 solve one boundary problem of fourth order with initial conditions, that appears, for example, when one solves the problem about lateral oscillations of elastic-viscous-relaxating rod of variable profile with variable momentum of inertia with freely supported ends.
{"title":"On solution of one linear problem with initial and boundary conditions","authors":"S. Kritskaia","doi":"10.15421/247729","DOIUrl":"https://doi.org/10.15421/247729","url":null,"abstract":"We solve one boundary problem of fourth order with initial conditions, that appears, for example, when one solves the problem about lateral oscillations of elastic-viscous-relaxating rod of variable profile with variable momentum of inertia with freely supported ends.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85149777","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 the paper, it is proved that$$1 - frac{1}{2n} leqslant suplimits_{substack{f in Cf ne const}} frac{E_n(f)_C}{omega_2(f; pi/n)_C} leqslant inflimits_{L_n in Z_n(C)} suplimits_{substack{f in Cf ne const}} frac{| f - L_n(f) |_C}{omega_2 (f; pi/n)_C} leqslant 1$$where $omega_2(f; t)_C$ is the modulus of smoothness of the function $f in C$, $E_n(f)_C$ is the best approximation by trigonometric polynomials of the degree not greater than $n-1$ in uniform metric, $Z_n(C)$ is the set of linear bounded operators that map $C$ to the subspace of trigonometric polynomials of degree not greater than $n-1$.
证明了$$1 - frac{1}{2n} leqslant suplimits_{substack{f in Cf ne const}} frac{E_n(f)_C}{omega_2(f; pi/n)_C} leqslant inflimits_{L_n in Z_n(C)} suplimits_{substack{f in Cf ne const}} frac{| f - L_n(f) |_C}{omega_2 (f; pi/n)_C} leqslant 1$$其中$omega_2(f; t)_C$是函数$f in C$的光滑模，$E_n(f)_C$是一致度规中不大于$n-1$次的三角多项式的最佳逼近，$Z_n(C)$是将$C$映射到不大于$n-1$次的三角多项式的子空间的线性有界算子的集合。
{"title":"To the question of approximation of continuous periodic functions by trigonometric polynomials","authors":"V. Shalaev","doi":"10.15421/247711","DOIUrl":"https://doi.org/10.15421/247711","url":null,"abstract":"In the paper, it is proved that$$1 - frac{1}{2n} leqslant suplimits_{substack{f in Cf ne const}} frac{E_n(f)_C}{omega_2(f; pi/n)_C} leqslant inflimits_{L_n in Z_n(C)} suplimits_{substack{f in Cf ne const}} frac{| f - L_n(f) |_C}{omega_2 (f; pi/n)_C} leqslant 1$$where $omega_2(f; t)_C$ is the modulus of smoothness of the function $f in C$, $E_n(f)_C$ is the best approximation by trigonometric polynomials of the degree not greater than $n-1$ in uniform metric, $Z_n(C)$ is the set of linear bounded operators that map $C$ to the subspace of trigonometric polynomials of degree not greater than $n-1$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81370803","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 obtain the solution of the problem about lateral oscillations of elastic-viscous-relaxating rod of finite length, variable profile, and variable momentum of rotation inertia of rod elements about the axis that is perpendicular to oscillation plane.
得到了有限长变截面弹性粘松弛杆和杆元转动惯量变动量沿垂直于振动面轴的横向振动问题的解。
{"title":"On one problem, associated with lateral oscillations of elastic-viscous-relaxating rod","authors":"D. Rogach","doi":"10.15421/247734","DOIUrl":"https://doi.org/10.15421/247734","url":null,"abstract":"We obtain the solution of the problem about lateral oscillations of elastic-viscous-relaxating rod of finite length, variable profile, and variable momentum of rotation inertia of rod elements about the axis that is perpendicular to oscillation plane.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75486091","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 a theorem on differential inequalities related to limit Cauchy problem for the set of ordinary differential equations$$y' = f(x,y,z),$$z' = varphi(x,y,z)$$with boundary conditions$$limlimits_{x rightarrow infty} y(x) = y(infty) = y_0, ; limlimits_{x rightarrow infty} z(x) = z(infty) = z_0$$
{"title":"On differential inequalities of S.A. Chaplygin related to limit Cauchy problem for sets of ordinary differential equations of first order","authors":"I. I. Bezvershenko","doi":"10.15421/247723","DOIUrl":"https://doi.org/10.15421/247723","url":null,"abstract":"We prove a theorem on differential inequalities related to limit Cauchy problem for the set of ordinary differential equations$$y' = f(x,y,z),$$z' = varphi(x,y,z)$$with boundary conditions$$limlimits_{x rightarrow infty} y(x) = y(infty) = y_0, ; limlimits_{x rightarrow infty} z(x) = z(infty) = z_0$$","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75439805","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 point out that$$inflimits_{L in L_n} suplimits_{substack{f in C_{2pi}f ne const}} frac{max | f(x) - L(f, x) |}{omega^*_2(f, pi/n + 1)} = frac{1}{2}$$where $C_{2pi}$ is the space of periodic continuous functions on real domain, $L_n$ is the set of linear operators that map $C_{2pi}$ to the set of trigonometric polynomials of order no greater than $n$ ($n = 0,1,ldots$), $omega_2(f, t) = suplimits_{x, |h| leqslant t} |f(x-h) - 2f(x) + f(x+h)|$, $omega^*_2(f, t)$ is the concave hull of the function $omega_2(f, t)$. In this equality, the infimum is attained for Korovkin's means.
{"title":"On one extremal property of Korovkin's means","authors":"V. Babenko, S. Pichugov","doi":"10.15421/247702","DOIUrl":"https://doi.org/10.15421/247702","url":null,"abstract":"We point out that$$inflimits_{L in L_n} suplimits_{substack{f in C_{2pi}f ne const}} frac{max | f(x) - L(f, x) |}{omega^*_2(f, pi/n + 1)} = frac{1}{2}$$where $C_{2pi}$ is the space of periodic continuous functions on real domain, $L_n$ is the set of linear operators that map $C_{2pi}$ to the set of trigonometric polynomials of order no greater than $n$ ($n = 0,1,ldots$), $omega_2(f, t) = suplimits_{x, |h| leqslant t} |f(x-h) - 2f(x) + f(x+h)|$, $omega^*_2(f, t)$ is the concave hull of the function $omega_2(f, t)$. In this equality, the infimum is attained for Korovkin's means.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74790522","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 obtain the solution of the problem about longitudinal impact of heterogeneous rod in presence of linear law of relaxation and after-effect in case when one end of rod is connected to some mass, and the other undergoes the impact by some load, we conduct the research of this solution.
{"title":"Research of solution of the problem about longitudinal oscillation of heterogeneous rod in presence of linear law of relaxation and after-effect","authors":"S. Kritskaia","doi":"10.15421/247730","DOIUrl":"https://doi.org/10.15421/247730","url":null,"abstract":"We obtain the solution of the problem about longitudinal impact of heterogeneous rod in presence of linear law of relaxation and after-effect in case when one end of rod is connected to some mass, and the other undergoes the impact by some load, we conduct the research of this solution.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75077149","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 the paper, we have found the supremum of the best mean approximations by algebraic polynomials of differentiable functions from $W^r_L$ classes for $r=1,2$.
本文在$r=1,2$的情况下，得到了$W^r_L$类中可微函数的代数多项式的最佳均值逼近的最优性。
{"title":"Exact values of the best mean approximations by algebraic polynomials of $W^r_L$ classes ($r=1,2$)","authors":"V. Kofanov","doi":"10.15421/247705","DOIUrl":"https://doi.org/10.15421/247705","url":null,"abstract":"In the paper, we have found the supremum of the best mean approximations by algebraic polynomials of differentiable functions from $W^r_L$ classes for $r=1,2$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74320094","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 establish a Tauberian theorem in the case of absolute summability in degree $p$ of double series by matrix methods, give its application to Abel methods.
{"title":"Tauberian theorems in the case of absolute summability in degree $p$ of double series","authors":"T.N. Yarkovaia","doi":"10.15421/247722","DOIUrl":"https://doi.org/10.15421/247722","url":null,"abstract":"We establish a Tauberian theorem in the case of absolute summability in degree $p$ of double series by matrix methods, give its application to Abel methods.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79140084","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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