Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,ldots,x_n)$ over $C$. Let $p,qin R$ be such that $pF^2(u)u+F^2(u)uq=0$ for all $uin S$. Then for all $xin R$ one of the followings holds:1) there exists $ain Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$,2) $p=-qin C$,3) $f(x_1,ldots,x_n)^2$ is central valued on $R$ and there exists $ain Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$.
{"title":"Generalized derivations of order $2$ on multilinear polynomials in prime rings","authors":"B. Prajapati, C. Gupta","doi":"10.30970/ms.58.1.26-35","DOIUrl":"https://doi.org/10.30970/ms.58.1.26-35","url":null,"abstract":"Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,ldots,x_n)$ over $C$. Let $p,qin R$ be such that \u0000$pF^2(u)u+F^2(u)uq=0$ for all $uin S$. \u0000Then for all $xin R$ one of the followings holds:1) there exists $ain Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$,2) $p=-qin C$,3) $f(x_1,ldots,x_n)^2$ is central valued on $R$ and there exists $ain Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41714504","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}
M. Pratsiovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak
In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2={frac12,1}$, $a_nin A_2$ and establish the normal property of numbers of the segment $I=[frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{sumlimits_{n=1}^{infty}(2a_n-1)v_n},$where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1in R$).
{"title":"Continued $mathbf{A_2}$-fractions and singular functions","authors":"M. Pratsiovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak","doi":"10.30970/ms.58.1.3-12","DOIUrl":"https://doi.org/10.30970/ms.58.1.3-12","url":null,"abstract":"In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2={frac12,1}$, $a_nin A_2$ and establish the normal property of numbers of the segment $I=[frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{sumlimits_{n=1}^{infty}(2a_n-1)v_n},$where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1in R$).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43923817","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}
Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the $C$-compactness is equivalent to the $AM$-compactness. Next we prove that, under mild assumptions, every linear section of a $C$-compact orthogonally additive operator is $AM$-compact, and every linear section of a narrow orthogonally additive operator is narrow.
{"title":"On linear sections of orthogonally additive operators","authors":"A. Gumenchuk, I. Krasikova, M. Popov","doi":"10.30970/ms.58.1.94-102","DOIUrl":"https://doi.org/10.30970/ms.58.1.94-102","url":null,"abstract":"Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the $C$-compactness is equivalent to the $AM$-compactness. Next we prove that, under mild assumptions, every linear section of a $C$-compact orthogonally additive operator is $AM$-compact, and every linear section of a narrow orthogonally additive operator is narrow.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48374781","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}
One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $Sigma$-norms of a group. A $Sigma$-norm is the intersection of the normalizers of all subgroups of a system $Sigma$. The authors study non-periodic groups with the restrictions on such a $Sigma$-norm -- the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{bar{p}})$of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{bar{p}})$.Additionally the relations between the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order and the norm $N_{G}(C_{infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{infty})$ of infinite cyclic subgroups norms $N_{G}(C _{infty})$ and $N_{G}(C _{bar{p}})$ coincide if and only if $N_{G}(C _{infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.In this case $N_{G}(C_{bar {p}})=N_{G}(C_{infty})=G$.
{"title":"Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order","authors":"M. Drushlyak, T. Lukashova","doi":"10.30970/ms.58.1.36-44","DOIUrl":"https://doi.org/10.30970/ms.58.1.36-44","url":null,"abstract":"One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $Sigma$-norms of a group. A $Sigma$-norm is the intersection of the normalizers of all subgroups of a system $Sigma$. The authors study non-periodic groups with the restrictions on such a $Sigma$-norm -- the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{bar{p}})$of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{bar{p}})$.Additionally the relations between the norm $N_{G}(C_{bar{p}})$ of cyclic subgroups of non-prime order and the norm $N_{G}(C_{infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{infty})$ of infinite cyclic subgroups norms $N_{G}(C _{infty})$ and $N_{G}(C _{bar{p}})$ coincide if and only if $N_{G}(C _{infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.In this case $N_{G}(C_{bar {p}})=N_{G}(C_{infty})=G$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46189507","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 extend some aspects of the essential spectra theory of linear operators acting in non-Archimedean (or p-adic) Banach spaces. In particular, we establish sufficient conditions for the relations between the essential spectra of the sum of two bounded linear operators and the union of their essential spectra. Moreover, we give essential prerequisites by studying the duality between p-adic upper and p-adic lower semi-Fredholm operators. We close this paper by giving some properties of the essential spectra.
{"title":"Essential spectra in non-Archimedean fields","authors":"A. Ammar, F. Boutaf, A. Jeribi","doi":"10.30970/ms.58.1.82-93","DOIUrl":"https://doi.org/10.30970/ms.58.1.82-93","url":null,"abstract":"In the paper we extend some aspects of the essential spectra theory of linear operators acting in non-Archimedean (or p-adic) Banach spaces. In particular, we establish sufficient conditions for the relations between the essential spectra of the sum of two bounded linear operators and the union of their essential spectra. Moreover, we give essential prerequisites by studying the duality between p-adic upper and p-adic lower semi-Fredholm operators. We close this paper by giving some properties of the essential spectra.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42124923","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}
Pub Date : 2022-04-16DOI: 10.30970/ms.59.2.141-155
R. Salimov, E. Sevost’yanov, V. Targonskii
The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation of the order~$p.$ Weseparately considered the situations of homeomorphisms and mappingswith branch points. In particular, we have established thathomeomorphisms of the Sobolev class satisfy the upper estimate ofthe distortion of the modulus at the inner and boundary points ofthe domain. In addition, we have proved that similar estimates ofcapacity distortion occur at the inner points of the domain for opendiscrete mappings. Also, we have shown that open discrete and closedmappings satisfy some estimates of the distortion of the modulus offamilies of paths at the boundary points. The results of themanuscript are obtained mainly under the condition that theso-called inner dilatation of mappings is locally integrable. Themain approach used in the proofs is the choice of admissiblefunctions, using the relations between the modulus and capacity, andconnections between different modulus of families of paths (similarto Hesse, Ziemer and Shlyk equalities). In this context, we haveobtained some lower estimate of the modulus of families of paths inSobolev classes. The manuscript contains some examples related toapplications of obtained results to specific mappings.
{"title":"On modulus inequality of the order $p$ for the inner dilatation","authors":"R. Salimov, E. Sevost’yanov, V. Targonskii","doi":"10.30970/ms.59.2.141-155","DOIUrl":"https://doi.org/10.30970/ms.59.2.141-155","url":null,"abstract":"The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation of the order~$p.$ Weseparately considered the situations of homeomorphisms and mappingswith branch points. In particular, we have established thathomeomorphisms of the Sobolev class satisfy the upper estimate ofthe distortion of the modulus at the inner and boundary points ofthe domain. In addition, we have proved that similar estimates ofcapacity distortion occur at the inner points of the domain for opendiscrete mappings. Also, we have shown that open discrete and closedmappings satisfy some estimates of the distortion of the modulus offamilies of paths at the boundary points. The results of themanuscript are obtained mainly under the condition that theso-called inner dilatation of mappings is locally integrable. Themain approach used in the proofs is the choice of admissiblefunctions, using the relations between the modulus and capacity, andconnections between different modulus of families of paths (similarto Hesse, Ziemer and Shlyk equalities). In this context, we haveobtained some lower estimate of the modulus of families of paths inSobolev classes. The manuscript contains some examples related toapplications of obtained results to specific mappings.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42113188","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 $ngeq 2$ and a real Banach space $E,$ ${mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$Pi(E)=Big{[x^*, (x_1, ldots, x_n)]: x^{*}(x_j)=|x^{*}|=|x_j|=1~mbox{for}~{j=1, ldots, n}Big}.$$For $Tin {mathcal L}(^n E:E),$ we define $$qopnamerelax o{Nr}({T})=Big{[x^*, (x_1, ldots, x_n)]in Pi(E): |x^{*}(T(x_1, ldots, x_n))|=v(T)Big},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {em numerical radius peak mapping} if there is $[x^{*}, (x_1, ldots, x_n)]in Pi(E)$ such that $qopnamerelax o{Nr}({T})={pm [x^{*}, (x_1, ldots, x_n)]}.$In this paper, we investigate some class of numerical radius peak mappings in ${mathcalL}(^n l_p:l_p)$ for $1leq p0.$Define $Tin {mathcal L}(^n l_p:l_p)$ by$$TBig(sum_{iin mathbb{N}}x_i^{(1)}e_i, cdots, sum_{iin mathbb{N}}x_i^{(n)}e_i Big)=sum_{jin mathbb{N}}a_{j}~x_{j}^{(1)}cdots x_{j}^{(n)}~e_j.qquadeqno(*)$$In particular is proved the following statements:$1.$ If $1< p<+infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0in mathbb{N}$ such that$$|a_{j_0}|>|a_{j}|~mbox{for every}~jin mathbb{N}backslash{j_0}.$$ $2.$ If $p=1$ then $T$ is not a numerical radius peak mapping in ${mathcal L}(^n l_1:l_1).$
{"title":"Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces","authors":"S. Kim","doi":"10.30970/ms.57.1.10-15","DOIUrl":"https://doi.org/10.30970/ms.57.1.10-15","url":null,"abstract":"For $ngeq 2$ and a real Banach space $E,$ ${mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$Pi(E)=Big{[x^*, (x_1, ldots, x_n)]: x^{*}(x_j)=|x^{*}|=|x_j|=1~mbox{for}~{j=1, ldots, n}Big}.$$For $Tin {mathcal L}(^n E:E),$ we define $$qopnamerelax o{Nr}({T})=Big{[x^*, (x_1, ldots, x_n)]in Pi(E): |x^{*}(T(x_1, ldots, x_n))|=v(T)Big},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {em numerical radius peak mapping} if there is $[x^{*}, (x_1, ldots, x_n)]in Pi(E)$ such that $qopnamerelax o{Nr}({T})={pm [x^{*}, (x_1, ldots, x_n)]}.$In this paper, we investigate some class of numerical radius peak mappings in ${mathcalL}(^n l_p:l_p)$ for $1leq p<infty.$ Let $(a_{j})_{jin mathbb{N}}$ be a bounded sequence in $mathbb{R}$ such that $sup_{jin mathbb{N}}|a_j|>0.$Define $Tin {mathcal L}(^n l_p:l_p)$ by$$TBig(sum_{iin mathbb{N}}x_i^{(1)}e_i, cdots, sum_{iin mathbb{N}}x_i^{(n)}e_i Big)=sum_{jin mathbb{N}}a_{j}~x_{j}^{(1)}cdots x_{j}^{(n)}~e_j.qquadeqno(*)$$In particular is proved the following statements:$1.$ If $1< p<+infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0in mathbb{N}$ such that$$|a_{j_0}|>|a_{j}|~mbox{for every}~jin mathbb{N}backslash{j_0}.$$ \u0000$2.$ If $p=1$ then $T$ is not a numerical radius peak mapping in ${mathcal L}(^n l_1:l_1).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48763694","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}
Let $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice ${z^0+tmathbf{b}: tinmathbb{C}}$ with the unit ball $mathbb{B}^n={zinmathbb{C}^: |z|:=sqrt{|z|_1^2+ldots+|z_n|^2}<1}$ for any $z^0inmathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $mathbf{b},$ where $mathbf{L}: mathbb{B}^ntomathbb{R}_+$ is a positive continuous function such that $L(z)>frac{beta|mathbf{b}|}{1-|z|}$ and $beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.
{"title":"Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties","authors":"Andriy Ivanovych Bandura, T. Salo, O. Skaskiv","doi":"10.30970/ms.57.1.68-78","DOIUrl":"https://doi.org/10.30970/ms.57.1.68-78","url":null,"abstract":"Let $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice ${z^0+tmathbf{b}: tinmathbb{C}}$ with the unit ball $mathbb{B}^n={zinmathbb{C}^: |z|:=sqrt{|z|_1^2+ldots+|z_n|^2}<1}$ for any $z^0inmathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $mathbf{b},$ where $mathbf{L}: mathbb{B}^ntomathbb{R}_+$ is a positive continuous function such that $L(z)>frac{beta|mathbf{b}|}{1-|z|}$ and $beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41822251","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 extent $e(X)$ of a topological space $X$ is the supremum of sizes of closed discrete subspaces of $X$. Assuming that $X$ belongs to some class of topological spaces, we bound $e(X)$ byother cardinal characteristics of $X$, for instance Lindel"of number, spread or density.
{"title":"Bounds on the extent of a topological space","authors":"A. Ravsky, T. Banakh","doi":"10.30970/ms.57.1.62-67","DOIUrl":"https://doi.org/10.30970/ms.57.1.62-67","url":null,"abstract":"The extent $e(X)$ of a topological space $X$ is the supremum of sizes of closed discrete subspaces of $X$. Assuming that $X$ belongs to some class of topological spaces, we bound $e(X)$ byother cardinal characteristics of $X$, for instance Lindel\"of number, spread or density.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45828192","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 present work, we study problem related to the approximation of continuous $2pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others. The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.
{"title":"An exact constant on the estimation of the approximation of classes of periodic functions of two variables by Ceśaro means","authors":"O. Rovenska","doi":"10.30970/ms.57.1.3-9","DOIUrl":"https://doi.org/10.30970/ms.57.1.3-9","url":null,"abstract":"In the present work, we study problem related to the approximation of continuous $2pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others. \u0000The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47765626","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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