Pub Date : 2023-06-30DOI: 10.15330/cmp.15.1.286-294
T. Stepaniuk, Yu. I. Kharkevych
Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $Lambda={lambda_{delta}(k)}$ of functions of the natural argument depending on the real parameter $delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.
{"title":"Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity","authors":"T. Stepaniuk, Yu. I. Kharkevych","doi":"10.15330/cmp.15.1.286-294","DOIUrl":"https://doi.org/10.15330/cmp.15.1.286-294","url":null,"abstract":"Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $Lambda={lambda_{delta}(k)}$ of functions of the natural argument depending on the real parameter $delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89681063","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 : 2023-06-30DOI: 10.15330/cmp.15.1.278-285
Z. Novosad
We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.
{"title":"Topological transitivity of translation operators in a non-separable Hilbert space","authors":"Z. Novosad","doi":"10.15330/cmp.15.1.278-285","DOIUrl":"https://doi.org/10.15330/cmp.15.1.278-285","url":null,"abstract":"We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79747647","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 : 2023-06-30DOI: 10.15330/cmp.15.1.295-305
A. Zhuchok, G. Pilz
We give new models of the free abelian dimonoid of rank $2$, the free generalized digroup of rank $1$ and the free commutative doppelsemigroup of rank $1$.
给出了秩为$2的自由阿贝尔二似子、秩为$1的自由广义二群和秩为$1的自由交换重半群的新模型。
{"title":"New models for some free algebras of small ranks","authors":"A. Zhuchok, G. Pilz","doi":"10.15330/cmp.15.1.295-305","DOIUrl":"https://doi.org/10.15330/cmp.15.1.295-305","url":null,"abstract":"We give new models of the free abelian dimonoid of rank $2$, the free generalized digroup of rank $1$ and the free commutative doppelsemigroup of rank $1$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"61 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89155173","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 : 2023-06-30DOI: 10.15330/cmp.15.1.306-314
A. Kuryliak, O. Skaskiv
Let $f$ be an analytic function in ${z: |z|0)(forall kinmathbb{N})(forall lambdainmathbb{R})colon mathbf{E}(e^{lambda Z_k})leq e^{D lambda^2}$, and such that $(existsbeta>0)(exists n_0inmathbb{N})colon inflimits_{ngeq n_0}mathbf{E}|Z_n|^{-beta}<+infty.$ It is proved that for every $delta>0$ there exists a set $E(delta)subset [0,R)$ of finite $h$-logarithmic measure (i.e. $intnolimits_{E}h(r)dln r<+infty$) such that almost surely for all $rin(r_0(omega),R)backslash E$ we have [ M_f(r,omega):=maxbig{|f(z,omega)|colon |z|=rbig}leq sqrt{h(r)}mu_f(r)Big(ln^3h(r)ln{h(r)mu_f(r)}Big)^{1/4+delta}, ] where $h(r)$ is any fixed continuous non-decreasing function on $[0;R)$ such that $h(r)geq2$ for all $rin (0,R)$ and $int^R_{r_{0}} h(r) dln r =+infty$ for some $r_0in(0,R)$.
{"title":"Sub-Gaussian random variables and Wiman's inequality for analytic functions","authors":"A. Kuryliak, O. Skaskiv","doi":"10.15330/cmp.15.1.306-314","DOIUrl":"https://doi.org/10.15330/cmp.15.1.306-314","url":null,"abstract":"Let $f$ be an analytic function in ${z: |z|<R}$ of the form $f(z)=sumlimits_{n=0}^{+infty}a_n z^n$. In the paper, we consider the Wiman-type inequality for random analytic functions of the form $f(z,omega)=sumlimits_{n=0}^{+infty}Z_n(omega)a_nz^n$, where $(Z_n)$ is a sequence on the Steinhaus probability space of real independent centered sub-Gaussian random variables, i.e. $(exists D>0)(forall kinmathbb{N})(forall lambdainmathbb{R})colon mathbf{E}(e^{lambda Z_k})leq e^{D lambda^2}$, and such that $(existsbeta>0)(exists n_0inmathbb{N})colon inflimits_{ngeq n_0}mathbf{E}|Z_n|^{-beta}<+infty.$ \u0000It is proved that for every $delta>0$ there exists a set $E(delta)subset [0,R)$ of finite $h$-logarithmic measure (i.e. $intnolimits_{E}h(r)dln r<+infty$) such that almost surely for all $rin(r_0(omega),R)backslash E$ we have [ M_f(r,omega):=maxbig{|f(z,omega)|colon |z|=rbig}leq sqrt{h(r)}mu_f(r)Big(ln^3h(r)ln{h(r)mu_f(r)}Big)^{1/4+delta}, ] where $h(r)$ is any fixed continuous non-decreasing function on $[0;R)$ such that $h(r)geq2$ for all $rin (0,R)$ and $int^R_{r_{0}} h(r) dln r =+infty$ for some $r_0in(0,R)$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78397724","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 : 2023-06-30DOI: 10.15330/cmp.15.1.270-277
M.L. Lourenço, V. Miranda
The main purpose of this paper is to study some geometric and topological properties of $c_0$-sum of the finite dimensional Banach lattice $ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattice $c_0(ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{infty}(ell_2^n)$ is an almost limited set.
{"title":"A note on the Banach lattice $c_0( ell_2^n)$, its dual and its bidual","authors":"M.L. Lourenço, V. Miranda","doi":"10.15330/cmp.15.1.270-277","DOIUrl":"https://doi.org/10.15330/cmp.15.1.270-277","url":null,"abstract":"The main purpose of this paper is to study some geometric and topological properties of $c_0$-sum of the finite dimensional Banach lattice $ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattice $c_0(ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{infty}(ell_2^n)$ is an almost limited set.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88427488","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 : 2023-06-29DOI: 10.15330/cmp.15.1.212-221
H. Karsli
This paper deals with construction and studying wavelet type Bernstein operators by using the compactly supported Daubechies wavelets of the given function $f$. The basis used in this construction is the wavelet expansion of the function $f$ instead of its rational sampling values $fbig( frac{k}{n}big)$. After that, we investigate some properties of these operators in some function spaces.
{"title":"On wavelet type Bernstein operators","authors":"H. Karsli","doi":"10.15330/cmp.15.1.212-221","DOIUrl":"https://doi.org/10.15330/cmp.15.1.212-221","url":null,"abstract":"This paper deals with construction and studying wavelet type Bernstein operators by using the compactly supported Daubechies wavelets of the given function $f$. The basis used in this construction is the wavelet expansion of the function $f$ instead of its rational sampling values $fbig( frac{k}{n}big)$. After that, we investigate some properties of these operators in some function spaces.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"108 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84910993","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 : 2023-06-29DOI: 10.15330/cmp.15.1.236-245
M. Sarıgöl
Using the uniform boundedness principle of Maddox, we characterize matrix transformations from the space $(ell_{p}) _{T}$ to the spaces $m(phi )$ and $n(phi )$ for the case $1leq pleq infty$, which correspond to bounded linear operators. Here $(ell _{p})_{T}$ is the domain of an arbitrary triangle matrix $T$ in the space $ell _{p}$, and the spaces $m(phi )$ and $n(phi )$ are introduced by W.L.C. Sargent. In special cases, we get some well known results of W.L.C. Sargent, M. Stieglitz and H. Tietz, E. Malkowsky and E. Savaş. Also we give other applications including some important new classes.
{"title":"Applications of uniform boundedness principle to matrix transformations","authors":"M. Sarıgöl","doi":"10.15330/cmp.15.1.236-245","DOIUrl":"https://doi.org/10.15330/cmp.15.1.236-245","url":null,"abstract":"Using the uniform boundedness principle of Maddox, we characterize matrix transformations from the space $(ell_{p}) _{T}$ to the spaces $m(phi )$ and $n(phi )$ for the case $1leq pleq infty$, which correspond to bounded linear operators. Here $(ell _{p})_{T}$ is the domain of an arbitrary triangle matrix $T$ in the space $ell _{p}$, and the spaces $m(phi )$ and $n(phi )$ are introduced by W.L.C. Sargent. In special cases, we get some well known results of W.L.C. Sargent, M. Stieglitz and H. Tietz, E. Malkowsky and E. Savaş. Also we give other applications including some important new classes.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82521614","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 : 2023-06-29DOI: 10.15330/cmp.15.1.196-211
J. Hasanov, I. Ekincioğlu, C. Keskin
We study the maximal operator $M_{gamma}$ and the singular integral operator $A_{gamma}$, associated with the generalized shift operator. The generalized shift operators are associated with the Laplace-Bessel differential operator. Our analysis is based on two weighted inequalities for the maximal operator, singular integral operators, and their commutators, related to the Laplace-Bessel differential operator in generalized weighted $B$-Morrey spaces.
{"title":"A characterization for $B$-singular integral operator and its commutators on generalized weighted $B$-Morrey spaces","authors":"J. Hasanov, I. Ekincioğlu, C. Keskin","doi":"10.15330/cmp.15.1.196-211","DOIUrl":"https://doi.org/10.15330/cmp.15.1.196-211","url":null,"abstract":"We study the maximal operator $M_{gamma}$ and the singular integral operator $A_{gamma}$, associated with the generalized shift operator. The generalized shift operators are associated with the Laplace-Bessel differential operator. Our analysis is based on two weighted inequalities for the maximal operator, singular integral operators, and their commutators, related to the Laplace-Bessel differential operator in generalized weighted $B$-Morrey spaces.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77672161","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 : 2023-06-29DOI: 10.15330/cmp.15.1.180-195
Andriy Ivanovych Bandura, O. Mulyava, M. Sheremeta
Let $F(s)=sumlimits_{n=1}^{infty}a_nexp{slambda_n}$ and $F_j(s)=sumlimits_{n=1}^{infty}a_{n,j}exp{slambda_n},$ $j=overline{1,p},$ be Dirichlet series with exponents $0lelambda_nuparrow+infty,$ $ntoinfty,$ and the abscissas of absolutely convergence equal to $0$. The function $F$ is called Hadamard composition of the genus $mge 1$ of the functions $F_j$ if $a_n=P(a_{n,1},dots ,a_{n,p})$, where $$P(x_1,dots ,x_p)=sumlimits_{k_1+dots+k_p=m}c_{k_1dots, k_p}x_1^{k_1}cdots x_p^{k_p}$$ is a homogeneous polynomial of degree $m$. In terms of generalized orders and convergence classes the connection between the growth of the functions $F_j$ and the growth of the Hadamard composition $F$ of the genus $mge 1$ of $F_j$ is investigated. The pseudostarlikeness and pseudoconvexity of the Hadamard composition of the genus $mge 1$ are studied.
{"title":"On Dirichlet series similar to Hadamard compositions in half-plane","authors":"Andriy Ivanovych Bandura, O. Mulyava, M. Sheremeta","doi":"10.15330/cmp.15.1.180-195","DOIUrl":"https://doi.org/10.15330/cmp.15.1.180-195","url":null,"abstract":"Let $F(s)=sumlimits_{n=1}^{infty}a_nexp{slambda_n}$ and $F_j(s)=sumlimits_{n=1}^{infty}a_{n,j}exp{slambda_n},$ $j=overline{1,p},$ be Dirichlet series with exponents $0lelambda_nuparrow+infty,$ $ntoinfty,$ and the abscissas of absolutely convergence equal to $0$. The function $F$ is called Hadamard composition of the genus $mge 1$ of the functions $F_j$ if $a_n=P(a_{n,1},dots ,a_{n,p})$, where $$P(x_1,dots ,x_p)=sumlimits_{k_1+dots+k_p=m}c_{k_1dots, k_p}x_1^{k_1}cdots x_p^{k_p}$$ is a homogeneous polynomial of degree $m$. In terms of generalized orders and convergence classes the connection between the growth of the functions $F_j$ and the growth of the Hadamard composition $F$ of the genus $mge 1$ of $F_j$ is investigated. The pseudostarlikeness and pseudoconvexity of the Hadamard composition of the genus $mge 1$ are studied.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"17 2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84507582","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 : 2023-06-29DOI: 10.15330/cmp.15.1.222-235
V. Litovchenko
We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $alphain(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $mathcal{P}_alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $alpha$-wandering model of the Lévy particle in the Euclidean space $mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle "leaves" the space $mathbb{R}^3$.
{"title":"Local Polya fluctuations of Riesz gravitational fields and the Cauchy problem","authors":"V. Litovchenko","doi":"10.15330/cmp.15.1.222-235","DOIUrl":"https://doi.org/10.15330/cmp.15.1.222-235","url":null,"abstract":"We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $alphain(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $mathcal{P}_alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $alpha$-wandering model of the Lévy particle in the Euclidean space $mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle \"leaves\" the space $mathbb{R}^3$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87052104","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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