Pub Date : 2024-07-14DOI: 10.15330/cmp.16.2.367-378
A.R. Khan, F. Rubab
We compare two linear functionals that are negative on convex functions. Further, using Green's functions we give some new conditions for reversed Jensen-Steffensen and related inequalities to hold. Using Green's function we also give refinement of Levinson type generalization of reversed Jensen-Steffensen and related inequalities. The acquired results are then used for constructing mean-value theorems.
{"title":"Узагальнені обернені нерівності Єнсена-Штеффенсена та пов’язані нерівності","authors":"A.R. Khan, F. Rubab","doi":"10.15330/cmp.16.2.367-378","DOIUrl":"https://doi.org/10.15330/cmp.16.2.367-378","url":null,"abstract":"We compare two linear functionals that are negative on convex functions. Further, using Green's functions we give some new conditions for reversed Jensen-Steffensen and related inequalities to hold. Using Green's function we also give refinement of Levinson type generalization of reversed Jensen-Steffensen and related inequalities. The acquired results are then used for constructing mean-value theorems.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141650595","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 : 2024-07-08DOI: 10.15330/cmp.16.2.351-366
K. Pozharska, A. S. Romanyuk, V. Romanyuk
Exact-order estimates are obtained for the entropy numbers and several types of widths (Kolmogorov, linear, trigonometric and orthowidth) for the Sobolev and Nikol'skii-Besov classes of one and several variables in the space $B_{q,1}$, $1
在$B_{q,1}$, $1
{"title":"Widths and entropy numbers of the classes of periodic functions of one and several variables in the space $B_{q,1}$","authors":"K. Pozharska, A. S. Romanyuk, V. Romanyuk","doi":"10.15330/cmp.16.2.351-366","DOIUrl":"https://doi.org/10.15330/cmp.16.2.351-366","url":null,"abstract":"Exact-order estimates are obtained for the entropy numbers and several types of widths (Kolmogorov, linear, trigonometric and orthowidth) for the Sobolev and Nikol'skii-Besov classes of one and several variables in the space $B_{q,1}$, $1 <q< infty$. It is shown, that in the multivariate case, in contrast to the univariate, the obtained estimates differ in order from the corresponding estimates in the space $L_q$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667064","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 : 2024-06-06DOI: 10.15330/cmp.16.1.158-173
O. V. Fedunyk-Yaremchuk, S. B. Hembars’ka, I.A. Romanyuk, P. V. Zaderei
We obtained the exact order estimates of approximation of periodic functions of several variables from the Nikol'skii-Besov-type classes $B^{Omega}_{p,theta}$ by using their step hyperbolic Fourier sums in the space $B_{q,1}$. The norm in this space is stronger than the $L_q$-norm. In the considered situations, approximations by the mentioned Fourier sums realize the orders of the best approximations by polynomials with "numbers" of harmonics from the step hyperbolic cross. We also established the exact order estimates of the Kolmogorov, linear and trigonometric widths of classes $B^{Omega}_{p,theta}$ in the space $B_{q,1}$ for certain relations between the parameters $p$ and $q$.
{"title":"Апроксимаційні характеристики класів типу Нікольського-Бєсова періодичних функцій багатьох змінних у просторі $B_{q,1}$","authors":"O. V. Fedunyk-Yaremchuk, S. B. Hembars’ka, I.A. Romanyuk, P. V. Zaderei","doi":"10.15330/cmp.16.1.158-173","DOIUrl":"https://doi.org/10.15330/cmp.16.1.158-173","url":null,"abstract":"We obtained the exact order estimates of approximation of periodic functions of several variables from the Nikol'skii-Besov-type classes $B^{Omega}_{p,theta}$ by using their step hyperbolic Fourier sums in the space $B_{q,1}$. The norm in this space is stronger than the $L_q$-norm. In the considered situations, approximations by the mentioned Fourier sums realize the orders of the best approximations by polynomials with \"numbers\" of harmonics from the step hyperbolic cross. We also established the exact order estimates of the Kolmogorov, linear and trigonometric widths of classes $B^{Omega}_{p,theta}$ in the space $B_{q,1}$ for certain relations between the parameters $p$ and $q$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141377901","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 : 2024-06-06DOI: 10.15330/cmp.16.1.174-189
T. Vasylyshyn
In this work, we investigate algebras of symmetric and block-symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions for which the $p$th power of the absolute value is Lebesgue integrable, where $pin[1,+infty),$ and Lebesgue measurable essentially bounded functions on $[0,1]$. We show that spectra of Fréchet algebras of block-symmetric entire functions of bounded type on these spaces consist only of point-evaluation functionals. Also we construct algebraic bases of algebras of continuous block-symmetric polynomials on these spaces. We generalize the above-mentioned results to a wide class of algebras of symmetric entire functions.
{"title":"Algebras of symmetric and block-symmetric functions on spaces of Lebesgue measurable functions","authors":"T. Vasylyshyn","doi":"10.15330/cmp.16.1.174-189","DOIUrl":"https://doi.org/10.15330/cmp.16.1.174-189","url":null,"abstract":"In this work, we investigate algebras of symmetric and block-symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions for which the $p$th power of the absolute value is Lebesgue integrable, where $pin[1,+infty),$ and Lebesgue measurable essentially bounded functions on $[0,1]$. We show that spectra of Fréchet algebras of block-symmetric entire functions of bounded type on these spaces consist only of point-evaluation functionals. Also we construct algebraic bases of algebras of continuous block-symmetric polynomials on these spaces. We generalize the above-mentioned results to a wide class of algebras of symmetric entire functions.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141376460","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 : 2024-06-06DOI: 10.15330/cmp.16.1.148-157
F. Erduvan, R. Keskin
In this study, it is shown that the only balancing numbers which are concatenations of three repdigits are $204$ and $1189$. The proof depends on lower bounds for linear forms and some tools from Diophantine approximation.
{"title":"Збалансовані числа, які є конкатенацією трьох репдиджитів","authors":"F. Erduvan, R. Keskin","doi":"10.15330/cmp.16.1.148-157","DOIUrl":"https://doi.org/10.15330/cmp.16.1.148-157","url":null,"abstract":"In this study, it is shown that the only balancing numbers which are concatenations of three repdigits are $204$ and $1189$. The proof depends on lower bounds for linear forms and some tools from Diophantine approximation.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141379698","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 : 2024-05-24DOI: 10.15330/cmp.16.1.103-113
S. Al-Kaseasbeh, M. Al Tahan
The purpose of this paper is to introduce linear Diophantine anti-fuzzification of algebraic structures. In this regard, we define linear Diophantine anti-fuzzy (LDAF) substructures of a semigroup and discuss some of its properties. Moreover, we characterize semigroups in terms of LDAF-ideals and LDAF-bi-ideals. Finally, we apply the linear Diophantine anti-fuzzification to groups and find a relationship between LDAF-subgroups of a group and its LDF-subgroups.
{"title":"Characterizations of semigroups by their linear Diophantine anti-fuzzy bi-ideals","authors":"S. Al-Kaseasbeh, M. Al Tahan","doi":"10.15330/cmp.16.1.103-113","DOIUrl":"https://doi.org/10.15330/cmp.16.1.103-113","url":null,"abstract":"The purpose of this paper is to introduce linear Diophantine anti-fuzzification of algebraic structures. In this regard, we define linear Diophantine anti-fuzzy (LDAF) substructures of a semigroup and discuss some of its properties. Moreover, we characterize semigroups in terms of LDAF-ideals and LDAF-bi-ideals. Finally, we apply the linear Diophantine anti-fuzzification to groups and find a relationship between LDAF-subgroups of a group and its LDF-subgroups.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141102199","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 : 2024-05-24DOI: 10.15330/cmp.16.1.114-127
S. Hristova, M.I. Abbas
The main aim of the current paper is to be appropriately defined several types of Ulam stability for non-linear fractional differential equation with generalized proportional fractional derivative of Riemann-Liouville type. In the new definitions, the initial values of the solutions of the given equation and the corresponding inequality could not coincide but they have to be closed enough. Some sufficient conditions for three types of Ulam stability for the studied equations are obtained, namely Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some of them are applied to a fractional generalization of a biological model.
{"title":"Аналіз стабільності типу Улама для узагальнених диференціальних рівнянь з пропорційними дробовими похідними","authors":"S. Hristova, M.I. Abbas","doi":"10.15330/cmp.16.1.114-127","DOIUrl":"https://doi.org/10.15330/cmp.16.1.114-127","url":null,"abstract":"The main aim of the current paper is to be appropriately defined several types of Ulam stability for non-linear fractional differential equation with generalized proportional fractional derivative of Riemann-Liouville type. In the new definitions, the initial values of the solutions of the given equation and the corresponding inequality could not coincide but they have to be closed enough. Some sufficient conditions for three types of Ulam stability for the studied equations are obtained, namely Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some of them are applied to a fractional generalization of a biological model.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141099918","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 : 2024-05-13DOI: 10.15330/cmp.16.1.93-102
R. Khats
Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)rightarrowmathbb C$, satisfying $int_{0}^1 t^4 |f(t)|^2, dt<+infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(rho_k)_{kinmathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $big{rho_k^2sqrt{xrho_k}J_{-5/2}(xrho_k):kinmathbb Nbig}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(rho_k)_{kinmathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $sigmale 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.
{"title":"Completeness of the systems of Bessel functions of index $-5/2$","authors":"R. Khats","doi":"10.15330/cmp.16.1.93-102","DOIUrl":"https://doi.org/10.15330/cmp.16.1.93-102","url":null,"abstract":"Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)rightarrowmathbb C$, satisfying $int_{0}^1 t^4 |f(t)|^2, dt<+infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(rho_k)_{kinmathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $big{rho_k^2sqrt{xrho_k}J_{-5/2}(xrho_k):kinmathbb Nbig}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(rho_k)_{kinmathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $sigmale 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140983949","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. Zabolotskyi, T.M. Zabolotskyi, S. Tarasyuk, Yu.M. Hal
Let $u$ be a subharmonic in $mathbb{R}^m$, $mgeq 3$, function of the zero kind with Riesz measure $mu$ on negative axis $Ox_1$, $n(r,u)=muleft({xinmathbb{R}^m colon |x|leq r}right)$, [N(r,u)=(m-2)int_1^r n(t,u)/t^{m-1}dt,] $rho(r)$ is a proximate order, $rho(r)torho$ as $rto+infty$, $0
{"title":"Regular behavior of subharmonic in space functions of the zero kind","authors":"M. Zabolotskyi, T.M. Zabolotskyi, S. Tarasyuk, Yu.M. Hal","doi":"10.15330/cmp.16.1.84-92","DOIUrl":"https://doi.org/10.15330/cmp.16.1.84-92","url":null,"abstract":"Let $u$ be a subharmonic in $mathbb{R}^m$, $mgeq 3$, function of the zero kind with Riesz measure $mu$ on negative axis $Ox_1$, $n(r,u)=muleft({xinmathbb{R}^m colon |x|leq r}right)$, [N(r,u)=(m-2)int_1^r n(t,u)/t^{m-1}dt,] $rho(r)$ is a proximate order, $rho(r)torho$ as $rto+infty$, $0<rho<1$. We found the asymptotic of $u(x)$ as $|x|to+infty$ by the condition $N(r,u)=left(1+o(1)right)r^{rho(r)}$, $rto+infty$. We also investigated the inverse relationship between a regular growth of $u$ and a behavior of $N(r,u)$ as $rto+infty$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140987052","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 deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.
{"title":"Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis","authors":"Kachanovsky N.A","doi":"10.15330/cmp.16.1.61-83","DOIUrl":"https://doi.org/10.15330/cmp.16.1.61-83","url":null,"abstract":"We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140988798","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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