Pub Date : 2023-06-23DOI: 10.30970/ms.59.2.115-122
A. Gatalevych, V. Shchedryk
This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy? The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $mathfrak{A}$ and $mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $mathfrak{A}_2$ and $mathfrak{E}_2$ denote a rings of $2 times 2$ matrices over them. We prove that full nonsingular matrices from $mathfrak{A}_2$ are adequate in $mathfrak{A}_2$ and full singular matrices from $mathfrak{E}_2$ are adequate in the set of full matrices in $mathfrak{E}_2$.
{"title":"On adequacy of full matrices","authors":"A. Gatalevych, V. Shchedryk","doi":"10.30970/ms.59.2.115-122","DOIUrl":"https://doi.org/10.30970/ms.59.2.115-122","url":null,"abstract":"This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy? \u0000The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $mathfrak{A}$ and $mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $mathfrak{A}_2$ and $mathfrak{E}_2$ denote a rings of $2 times 2$ matrices over them. We prove that full nonsingular matrices from $mathfrak{A}_2$ are adequate in $mathfrak{A}_2$ and full singular matrices from $mathfrak{E}_2$ are adequate in the set of full matrices in $mathfrak{E}_2$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48758825","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 : 2023-06-23DOI: 10.30970/ms.59.2.156-167
Andriy Ivanovych Bandura, F. Nuray
Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+lin{0, 1, 2, ldots, M}$, for some integer $pge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:begin{gather*}sum_{i+j=0}^{M}frac{left(int_{0}^{2pi}int_{0}^{2pi}|f^{(i+k,j+l)}(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})|^{p}dtheta_{1}theta_{2}right)^{frac{1}{p}}}{i!j!}ge ge sum_{i+j=M+1}^{infty}frac{left(int_{0}^{2pi}int_{0}^{2pi}|f^{(i+k,j+l)}(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})|^{p}dtheta_{1}theta_{2}right)^{frac{1}{p}}}{i!j!},end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding[2+2logBig(1+frac{1}{C}Big)+log[(2M)!/M!].]If this condition is replaced by related conditions, then also $f$ is of exponential type.
{"title":"Entire Bivariate Functions of Exponential Type II","authors":"Andriy Ivanovych Bandura, F. Nuray","doi":"10.30970/ms.59.2.156-167","DOIUrl":"https://doi.org/10.30970/ms.59.2.156-167","url":null,"abstract":"Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+lin{0, 1, 2, ldots, M}$, for some integer $pge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:begin{gather*}sum_{i+j=0}^{M}frac{left(int_{0}^{2pi}int_{0}^{2pi}|f^{(i+k,j+l)}(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})|^{p}dtheta_{1}theta_{2}right)^{frac{1}{p}}}{i!j!}ge ge sum_{i+j=M+1}^{infty}frac{left(int_{0}^{2pi}int_{0}^{2pi}|f^{(i+k,j+l)}(r_{1}e^{mathbf{i}theta_{1}},r_{2}e^{mathbf{i}theta_{2}})|^{p}dtheta_{1}theta_{2}right)^{frac{1}{p}}}{i!j!},end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding[2+2logBig(1+frac{1}{C}Big)+log[(2M)!/M!].]If this condition is replaced by related conditions, then also $f$ is of exponential type.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49595530","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 : 2023-06-23DOI: 10.30970/ms.59.2.168-177
V. MatematychniStudii., No 59, R. S. Dyavanal, S. B. Kalakoti
The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.
{"title":"Normality and uniqueness of homogeneous differential polynomials","authors":"V. MatematychniStudii., No 59, R. S. Dyavanal, S. B. Kalakoti","doi":"10.30970/ms.59.2.168-177","DOIUrl":"https://doi.org/10.30970/ms.59.2.168-177","url":null,"abstract":"The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41732653","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 : 2023-06-23DOI: 10.30970/ms.59.2.187-200
S. Radchenko, V. Samoilenko, P. Samusenko
The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.
{"title":"Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients","authors":"S. Radchenko, V. Samoilenko, P. Samusenko","doi":"10.30970/ms.59.2.187-200","DOIUrl":"https://doi.org/10.30970/ms.59.2.187-200","url":null,"abstract":"The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42707806","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 : 2023-06-23DOI: 10.30970/ms.59.2.132-140
O. Mulyava, M. Sheremeta
A function $F(s)=sum_{n=1}^{infty}a_nexp{slambda_n}$ with $0lelambda_nuparrow+infty$ is called the Hadamard composition of the genus $mge 1$ of functions $F_j(s)=sum_{n=1}^{infty}a_{n,j}exp{slambda_n}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=sumlimits_{k_1+dots+k_p=m}c_{k_1...k_p}x_1^{k_1}cdot...cdot x_p^{k_p}$ is a homogeneous polynomial of degree $mge 1$. Let $M(sigma,F)=sup{|F(sigma+it)|:,tin{Bbb R}}$ and functions $alpha,,beta$ be positive continuous and increasing to $+infty$ on $[x_0, +infty)$. To characterize the growth of the function $M(sigma,F)$, we use generalized order $varrho_{alpha,beta}[F]=varlimsuplimits_{sigmato+infty}dfrac{alpha(ln,M(sigma,F))}{beta(sigma)}$, generalized type$T_{alpha,beta}[F]=varlimsuplimits_{sigmato+infty}dfrac{ln,M(sigma,F)}{alpha^{-1}(varrho_{alpha,beta}[F]beta(sigma))}$and membership in the convergence class defined by the condition$displaystyle int_{sigma_0}^{infty}frac{ln,M(sigma,F)}{sigmaalpha^{-1}(varrho_{alpha,beta}[F]beta(sigma))}dsigma<+infty.$Assuming the functions $alpha, beta$ and $alpha^{-1}(cbeta(ln,x))$ are slowly increasing for each $cin (0,+infty)$ and $ln,n=O(lambda_n)$ as $nto infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $varrho_{alpha,beta}[F_j]=varrhoin (0,+infty)$ and the types $T_{alpha,beta}[F_j]=T_jin [0,+infty)$, $c_{m0...0}=cnot=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $ntoinfty$ for $2le jle p$, and $F$ is the Hadamard composition of genus$mge 1$ of the functions $F_j$ then $varrho_{alpha,beta}[F]=varrho$ and $displaystyle T_{alpha,beta}[F]le sum_{k_1+dots+k_p=m}(k_1T_1+...+k_pT_p).$It is proved also that $F$ belongs to the generalized convergence class if and only ifall functions $F_j$ belong to the same convergence class.
{"title":"On entire Dirichlet series similar to Hadamard compositions","authors":"O. Mulyava, M. Sheremeta","doi":"10.30970/ms.59.2.132-140","DOIUrl":"https://doi.org/10.30970/ms.59.2.132-140","url":null,"abstract":"A function $F(s)=sum_{n=1}^{infty}a_nexp{slambda_n}$ with $0lelambda_nuparrow+infty$ is called the Hadamard composition of the genus $mge 1$ of functions $F_j(s)=sum_{n=1}^{infty}a_{n,j}exp{slambda_n}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=sumlimits_{k_1+dots+k_p=m}c_{k_1...k_p}x_1^{k_1}cdot...cdot x_p^{k_p}$ is a homogeneous polynomial of degree $mge 1$. Let $M(sigma,F)=sup{|F(sigma+it)|:,tin{Bbb R}}$ and functions $alpha,,beta$ be positive continuous and increasing to $+infty$ on $[x_0, +infty)$. To characterize the growth of the function $M(sigma,F)$, we use generalized order $varrho_{alpha,beta}[F]=varlimsuplimits_{sigmato+infty}dfrac{alpha(ln,M(sigma,F))}{beta(sigma)}$, generalized type$T_{alpha,beta}[F]=varlimsuplimits_{sigmato+infty}dfrac{ln,M(sigma,F)}{alpha^{-1}(varrho_{alpha,beta}[F]beta(sigma))}$and membership in the convergence class defined by the condition$displaystyle int_{sigma_0}^{infty}frac{ln,M(sigma,F)}{sigmaalpha^{-1}(varrho_{alpha,beta}[F]beta(sigma))}dsigma<+infty.$Assuming the functions $alpha, beta$ and $alpha^{-1}(cbeta(ln,x))$ are slowly increasing for each $cin (0,+infty)$ and $ln,n=O(lambda_n)$ as $nto infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $varrho_{alpha,beta}[F_j]=varrhoin (0,+infty)$ and the types $T_{alpha,beta}[F_j]=T_jin [0,+infty)$, $c_{m0...0}=cnot=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $ntoinfty$ for $2le jle p$, and $F$ is the Hadamard composition of genus$mge 1$ of the functions $F_j$ then $varrho_{alpha,beta}[F]=varrho$ and $displaystyle T_{alpha,beta}[F]le sum_{k_1+dots+k_p=m}(k_1T_1+...+k_pT_p).$It is proved also that $F$ belongs to the generalized convergence class if and only ifall functions $F_j$ belong to the same convergence class.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44846177","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 : 2023-05-16DOI: 10.30970/ms.59.2.201-204
O. Rovenska
The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials. In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.
{"title":"Approximation of classes of Poisson integrals by Fejer means","authors":"O. Rovenska","doi":"10.30970/ms.59.2.201-204","DOIUrl":"https://doi.org/10.30970/ms.59.2.201-204","url":null,"abstract":"The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. \u0000The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials. \u0000In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45728919","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 : 2023-05-13DOI: 10.30970/ms.59.2.205-214
I. Banakh, T. Banakh, Maria Kolinko, A. Ravsky
A metric space $(X,mathsf{d})$ is called a {em subline} if every 3-element subset $T$ of $X$ can be written as $T={x,y,z}$ for some points $x,y,z$ such that $mathsf{d}(x,z)=mathsf{d}(x,y)+mathsf{d}(y,z)$. By a classical result of Menger, every subline of cardinality $ne 4$ is isometric to a subspace of the real line. A subline $(X,mathsf{d})$ is called an {em $n$-subline} for a natural number $n$ if for every $cin X$ and positive real number $rinmathsf{d}[X^2]$, the sphere ${mathsf S}(c;r):={xin Xcolon mathsf{d}(x,c)=r}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $Gsubseteq{mathbb R}$, a metric space $(X,mathsf{d})$ is isometric to $G$ if and only if $X$ is a $2$-subline with $mathsf{d}[X^2]=G_+:= Gcap[0,infty)$. A metric space $(X,mathsf{d})$ is called a {em ray} if $X$ is a $1$-subline and $X$ contains a point $oin X$ such that for every $rinmathsf{d}[X^2]$ the sphere ${mathsf S}(o;r)$ is a singleton. We prove that for a subgroup $Gsubseteq{mathbb Q}$, a metric space $(X,mathsf{d})$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $mathsf{d}[X^2]=G_+$. A metric space $X$ is isometric to the ray ${mathbb R}_+$ if and only if $X$ is a complete ray such that ${mathbb Q}_+subseteq mathsf{d}[X^2]$. On the other hand, the real line contains a dense ray $Xsubseteq{mathbb R}$ such that $mathsf{d}[X^2]={mathbb R}_+$.
{"title":"Metric characterizations of some subsets of the real line","authors":"I. Banakh, T. Banakh, Maria Kolinko, A. Ravsky","doi":"10.30970/ms.59.2.205-214","DOIUrl":"https://doi.org/10.30970/ms.59.2.205-214","url":null,"abstract":"A metric space $(X,mathsf{d})$ is called a {em subline} if every 3-element subset $T$ of $X$ can be written as $T={x,y,z}$ for some points $x,y,z$ such that $mathsf{d}(x,z)=mathsf{d}(x,y)+mathsf{d}(y,z)$. By a classical result of Menger, every subline of cardinality $ne 4$ is isometric to a subspace of the real line. A subline $(X,mathsf{d})$ is called an {em $n$-subline} for a natural number $n$ if for every $cin X$ and positive real number $rinmathsf{d}[X^2]$, the sphere ${mathsf S}(c;r):={xin Xcolon mathsf{d}(x,c)=r}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $Gsubseteq{mathbb R}$, a metric space $(X,mathsf{d})$ is isometric to $G$ if and only if $X$ is a $2$-subline with $mathsf{d}[X^2]=G_+:= Gcap[0,infty)$. A metric space $(X,mathsf{d})$ is called a {em ray} if $X$ is a $1$-subline and $X$ contains a point $oin X$ such that for every $rinmathsf{d}[X^2]$ the sphere ${mathsf S}(o;r)$ is a singleton. We prove that for a subgroup $Gsubseteq{mathbb Q}$, a metric space $(X,mathsf{d})$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $mathsf{d}[X^2]=G_+$. A metric space $X$ is isometric to the ray ${mathbb R}_+$ if and only if $X$ is a complete ray such that ${mathbb Q}_+subseteq mathsf{d}[X^2]$. On the other hand, the real line contains a dense ray $Xsubseteq{mathbb R}$ such that $mathsf{d}[X^2]={mathbb R}_+$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47173806","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 $(lambda_n)_{n=0}^{+infty}$ be a nonnegative sequence increasing to $+infty$, $F(s)=sum_{n=0}^{+infty} a_ne^{slambda_n}$ be an absolutely convergent Dirichlet series in the half-plane ${sinmathbb{C}colon operatorname{Re} s<0}$, and let, for every $sigma<0$, $mathfrak{M}(sigma,F)=sum_{n=0}^{+infty} |a_n|e^{sigmalambda_n}$. Suppose that $Phicolon (-infty,0)tooverline{mathbb{R}}$ is a function, and let $widetilde{Phi}(x)$ be the Young-conjugate function of $Phi(sigma)$, i.e.$widetilde{Phi}(x)=sup{xsigma-alpha(sigma)colon sigma<0}$ for all $xinmathbb{R}$. In the article, the following two statements are proved: (i) There exist constants $thetain(0,1)$ and $Cinmathbb{R}$ such that$lnmathfrak{M}(sigma,F)lePhi(thetasigma)+C$ for all $sigma<0$ if and only if there exist constants $deltain(0, 1)$ and $cinmathbb{R}$ such that $lnsum_{m=0}^n|a_m|le-widetilde{Phi}(lambda_n/delta)+c$ for all integers $nge0$ (Theorem 2); (ii) For every $thetain(0,1)$ there exists a real constant $C=C(delta)$ such that $lnmathfrak{M}(sigma,F)lePhi( thetasigma)+C$ for all $sigma<0$ if and only if for every $deltain(0,1)$ there exists a real constant $c=c(delta)$ such that $lnsum_{m=0}^n|a_m|le-widetilde{Phi}(lambda_n/delta)+c$ for all integers $nge0$ (Theorem 3).iii) Let $Phi$ be a continuous positive increasing function on $mathbb{R}$ such that $Phi(sigma)/sigmato+infty$, $sigmato+ infty$ and $F$ be a entire Dirichlet series. For every $q>1$ there exists a constant $C=C(q)inmathbb{R}$ such that $lnmathfrak{M}(sigma,F)le Phi(qsigma)+C,quad sigmainmathbb{R},$ holds if and only if for every $delta in(0,1)$ there exist constants $c=c(delta)inmathbb{R}$ and $n_0=n_0(delta)inmathbb{N}_0$ such that $ln sum_{m=n}^{+infty}|a_m|le-widetilde{Phi}(deltalambda_n)+c,quad nge n_0$ Theorem 5. These results are analogous to some results previously obtained by M.M. Sheremeta for entire Dirichlet series.
{"title":"Global estimates for sums of absolutely convergent Dirichlet series in a half-plane","authors":"P. Filevych, O. B. Hrybel","doi":"10.30970/ms.59.1.60-67","DOIUrl":"https://doi.org/10.30970/ms.59.1.60-67","url":null,"abstract":"Let $(lambda_n)_{n=0}^{+infty}$ be a nonnegative sequence increasing to $+infty$, $F(s)=sum_{n=0}^{+infty} a_ne^{slambda_n}$ be an absolutely convergent Dirichlet series in the half-plane ${sinmathbb{C}colon operatorname{Re} s<0}$, and let, for every $sigma<0$, $mathfrak{M}(sigma,F)=sum_{n=0}^{+infty} |a_n|e^{sigmalambda_n}$. \u0000Suppose that $Phicolon (-infty,0)tooverline{mathbb{R}}$ is a function, and let $widetilde{Phi}(x)$ be the Young-conjugate function of $Phi(sigma)$, i.e.$widetilde{Phi}(x)=sup{xsigma-alpha(sigma)colon sigma<0}$ for all $xinmathbb{R}$. In the article, the following two statements are proved: \u0000(i) There exist constants $thetain(0,1)$ and $Cinmathbb{R}$ such that$lnmathfrak{M}(sigma,F)lePhi(thetasigma)+C$ for all $sigma<0$ if and only if there exist constants $deltain(0, 1)$ and $cinmathbb{R}$ such that $lnsum_{m=0}^n|a_m|le-widetilde{Phi}(lambda_n/delta)+c$ for all integers $nge0$ (Theorem 2); \u0000(ii) For every $thetain(0,1)$ there exists a real constant $C=C(delta)$ such that $lnmathfrak{M}(sigma,F)lePhi( thetasigma)+C$ for all $sigma<0$ if and only if for every $deltain(0,1)$ there exists a real constant $c=c(delta)$ such that $lnsum_{m=0}^n|a_m|le-widetilde{Phi}(lambda_n/delta)+c$ for all integers $nge0$ (Theorem 3).iii) Let $Phi$ be a continuous positive increasing function on $mathbb{R}$ such that $Phi(sigma)/sigmato+infty$, $sigmato+ infty$ and $F$ be a entire Dirichlet series. \u0000For every $q>1$ there exists a constant $C=C(q)inmathbb{R}$ such that $lnmathfrak{M}(sigma,F)le Phi(qsigma)+C,quad sigmainmathbb{R},$ holds if and only if for every $delta in(0,1)$ there exist constants $c=c(delta)inmathbb{R}$ and $n_0=n_0(delta)inmathbb{N}_0$ such that $ln sum_{m=n}^{+infty}|a_m|le-widetilde{Phi}(deltalambda_n)+c,quad nge n_0$ Theorem 5. \u0000These results are analogous to some results previously obtained by M.M. Sheremeta for entire Dirichlet series.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46052251","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 : 2023-03-28DOI: 10.30970/ms.59.1.106-112
C. Santhoshkumar
In this paper, we give simple characterization of binormal weighted composition operators $C_{psi, phi}$ on the Fock space over $mathbb{C}$ where weight function is of the form $psi(zeta) = e^{langle zeta, c rangle}$ for some $c in mathbb{C}$. We derive conditions for $C_{phi}$ to be binormal such that $C^*_{phi}C_{phi}$ and $C^*_{phi} + C_{phi}$ commute. Finally we give some simple characterization of binormal weighted composition operator to be complex symmetric.
本文给出了在$mathbb{C}$上的Fock空间上的二正规加权复合算子$C_{psi, phi}$的简单刻画，其中对于某些$c in mathbb{C}$，权函数的形式为$psi(zeta) = e^{langle zeta, c rangle}$。我们推导出$C_{phi}$是异正规的条件，使得$C^*_{phi}C_{phi}$和$C^*_{phi} + C_{phi}$可以交换。最后给出了二正规加权复合算子复对称的一些简单刻画。
{"title":"Binormal and complex symmetric weighted composition operators on the Fock Space over $mathbb{C}$","authors":"C. Santhoshkumar","doi":"10.30970/ms.59.1.106-112","DOIUrl":"https://doi.org/10.30970/ms.59.1.106-112","url":null,"abstract":"In this paper, we give simple characterization of binormal weighted composition operators $C_{psi, phi}$ on the Fock space over $mathbb{C}$ where weight function is of the form $psi(zeta) = e^{langle zeta, c rangle}$ for some $c in mathbb{C}$. We derive conditions for $C_{phi}$ to be binormal such that $C^*_{phi}C_{phi}$ and $C^*_{phi} + C_{phi}$ commute. Finally we give some simple characterization of binormal weighted composition operator to be complex symmetric.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44873878","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 Banach and Fr'{e}chet spaces of series $A(z)=sum_{n=1}^{infty}a_nf(lambda_nz)$ regularly converging in ${mathbb C}$,where $f$ is an entire transcendental function and $(lambda_n)$ is a sequence of positive numbers increasing to $+infty$, are studied.Let $M_f(r)=max{|f(z)|:,|z|=r}$, $Gamma_f(r)=frac{dln,M_f(r)}{dln,r}$, $h$ be positive continuous function on $[0,+infty)$increasing to $+infty$ and ${bf S}_h(f,Lambda)$ be a class of the function $A$ such that $|a_n|M_f(lambda_nh(lambda_n))$ $to 0$ as$nto+infty$. Define $|A|_h=max{|a_n|M_f(lambda_nh(lambda_n)):nge 1}$. It is proved that if$ln,n=o(Gamma_f(lambda_n))$ as $ntoinfty$ then $({bf S}_h(f,Lambda),|cdot|_h)$ is a non-uniformly convexBanach space which is also separable.In terms of generalized orders, the relationship between the growth of $mathfrak{M}(r,A)=break=sum_{n=1}^{infty} |a_n|M_f(rlambda_n)$,the maximal term $mu(r,A)= max{|a_n|M_f(rlambda_n)colon nge 1}$ and the central index$nu(r,A)= max{nge 1colon |a_n|M_f(rlambda_n)=mu(r,A)}$ and the decrease of the coefficients $a_n$.The results obtained are used to construct Fr'{e}chet spaces of series in systems of functions.
{"title":"Spaces of series in system of functions","authors":"M. Sheremeta","doi":"10.30970/ms.59.1.46-59","DOIUrl":"https://doi.org/10.30970/ms.59.1.46-59","url":null,"abstract":"The Banach and Fr'{e}chet spaces of series $A(z)=sum_{n=1}^{infty}a_nf(lambda_nz)$ regularly converging in ${mathbb C}$,where $f$ is an entire transcendental function and $(lambda_n)$ is a sequence of positive numbers increasing to $+infty$, are studied.Let $M_f(r)=max{|f(z)|:,|z|=r}$, $Gamma_f(r)=frac{dln,M_f(r)}{dln,r}$, $h$ be positive continuous function on $[0,+infty)$increasing to $+infty$ and ${bf S}_h(f,Lambda)$ be a class of the function $A$ such that $|a_n|M_f(lambda_nh(lambda_n))$ $to 0$ as$nto+infty$. Define $|A|_h=max{|a_n|M_f(lambda_nh(lambda_n)):nge 1}$. It is proved that if$ln,n=o(Gamma_f(lambda_n))$ as $ntoinfty$ then $({bf S}_h(f,Lambda),|cdot|_h)$ is a non-uniformly convexBanach space which is also separable.In terms of generalized orders, the relationship between the growth of $mathfrak{M}(r,A)=break=sum_{n=1}^{infty} |a_n|M_f(rlambda_n)$,the maximal term $mu(r,A)= max{|a_n|M_f(rlambda_n)colon nge 1}$ and the central index$nu(r,A)= max{nge 1colon |a_n|M_f(rlambda_n)=mu(r,A)}$ and the decrease of the coefficients $a_n$.The results obtained are used to construct Fr'{e}chet spaces of series in systems of functions.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69301863","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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