In this paper, we obtain some reverses of Callebaut and Hölder inequalities for isotonic functionals via a reverse of Young’s inequality we have established recently. Applications for integrals and n-tuples of real numbers are provided as well.
在本文中，我们通过最近建立的杨氏不等式的逆定理，得到了等价函数的 Callebaut 和 Hölder 不等式的一些逆定理。本文还提供了积分和 n 个实数元组的应用。
{"title":"Some additive reverses of Callebaut and Hölder inequalities for isotonic functionals","authors":"S. Dragomir","doi":"10.33205/cma.1362691","DOIUrl":"https://doi.org/10.33205/cma.1362691","url":null,"abstract":"In this paper, we obtain some reverses of Callebaut and Hölder inequalities for isotonic functionals via a reverse of Young’s inequality we have established recently. Applications for integrals and n-tuples of real numbers are provided as well.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139221318","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 this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.
{"title":"On a new approach in the space of measurable functions","authors":"A. Aral","doi":"10.33205/cma.1381787","DOIUrl":"https://doi.org/10.33205/cma.1381787","url":null,"abstract":"In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"22 19","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139272188","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 this paper, we characterize the system of left translates ${L_{(2k,l,m)}g:k,l,minmathbb{Z}}$, $gin L^2(mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^lambda$. Here, $mathbb{H}$ denotes the Heisenberg group and $g^lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates ${L_{(2k,l,m)}g:k,l,minmathbb{Z}}$ on $mathbb{H}$. This result is also illustrated with an example.
在本文中，我们刻画了左平移${L_{(2k,l,m)}g:k,l,m在mathbb{Z}}$中，$g在l ^2(mathbb{H})$中，根据相应函数$g^lambda$的扭平移来表示的一个帧序列或Riesz序列。这里，$mathbb{H}$表示海森堡群，$g^ λ $表示$g$关于中心变量的傅里叶反变换。Riesz序列的这种特征使我们能够找到一些具体的例子。我们还研究了左平移系统${L_{(2k,l,m)}g:k,l,minmathbb{Z}}$ on $mathbb{H}$的斜对偶结构。最后给出了一个算例。
{"title":"Systems of left translates and oblique duals on the Heisenberg group","authors":"Santi DAS, Radha RAMAKRİSHNAN, Peter MASSOPUST","doi":"10.33205/cma.1382306","DOIUrl":"https://doi.org/10.33205/cma.1382306","url":null,"abstract":"In this paper, we characterize the system of left translates ${L_{(2k,l,m)}g:k,l,minmathbb{Z}}$, $gin L^2(mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^lambda$. Here, $mathbb{H}$ denotes the Heisenberg group and $g^lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates ${L_{(2k,l,m)}g:k,l,minmathbb{Z}}$ on $mathbb{H}$. This result is also illustrated with an example.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" 34","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292742","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 this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE-T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in Grünbaum's paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership.
{"title":"On the eigenvalue-separation properties of real tridiagonal matrices","authors":"Yan WU, Ludwig KOHAUPT","doi":"10.33205/cma.1330647","DOIUrl":"https://doi.org/10.33205/cma.1330647","url":null,"abstract":"In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE-T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in Grünbaum's paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136037736","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 this paper, we give some recent and new results for some contraction mappings on O−complete metric-like space and also we give illustrative examples. At the end, we give an application to show the existence of a solution of a differential equation.
{"title":"Some recent and new fixed point results on orthogonal metric-like space","authors":"Özlem ACAR","doi":"10.33205/cma.1360402","DOIUrl":"https://doi.org/10.33205/cma.1360402","url":null,"abstract":"In this paper, we give some recent and new results for some contraction mappings on O−complete metric-like space and also we give illustrative examples. At the end, we give an application to show the existence of a solution of a differential equation.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"202 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135352804","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}
Extensions of a positive hermitian linear functional $omega$, defined on a dense *-subalgebra $mathfrak{A_{0}}$ of a topological *-algebra $mathfrak{A}[tau]$ are analyzed. It turns out that their maximal extension as linear functionals or hermitian linear functional are everywhere defined. The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [1] is rivisited from this point of view. Examples mostly taken from the theory of integration are discussed.
{"title":"Maximal extensions of a linear functional","authors":"Fabio BURDERİ, Camillo TRAPANI, Salvatore TRİOLO","doi":"10.33205/cma.1310238","DOIUrl":"https://doi.org/10.33205/cma.1310238","url":null,"abstract":"Extensions of a positive hermitian linear functional $omega$, defined on a dense *-subalgebra $mathfrak{A_{0}}$ of a topological *-algebra $mathfrak{A}[tau]$ are analyzed. It turns out that their maximal extension as linear functionals or hermitian linear functional are everywhere defined. The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [1] is rivisited from this point of view. Examples mostly taken from the theory of integration are discussed.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135485984","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 work we estimate the $mathbb{L}^1$ norms of some special cosine and sine series used in studying fractional integrals.
在这项工作中，我们估计了一些用于研究分数积分的特殊余弦和正弦级数的$mathbb｛L｝^1$范数。
{"title":"Estimates of the norms of some cosine and sine series","authors":"J. Bustamante","doi":"10.33205/cma.1345440","DOIUrl":"https://doi.org/10.33205/cma.1345440","url":null,"abstract":"In the work we estimate the $mathbb{L}^1$ norms of some special cosine and sine series used in studying fractional integrals.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43817213","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}
V. Gutlyanski̇i̇, O. Nesmelova, V. Ryazanov, E. Yakubov
We study the semi-linear Beltrami equation $omega_{bar{z}}-mu(z) omega_z=sigma(z)q(omega(z))$ and show that it is closely related to the corresponding semi-linear equation of the form ${rm div} A(z)nabla,U(z)=G(z) Q(U(z)).$ Applying the theory of completely continuous operators by Ahlfors-Bers and Leray-Schauder, we prove existence of regular solutions both to the semi-linear Beltrami equation and to the given above semi-linear equation in the divergent form, see Theorems 1.1 and 5.2. We also derive their representation through solutions of the semi-linear Vekua type equations and generalized analytic functions with sources. Finally, we apply Theorem 5.2 for several model equations describing physical phenomena in anisotropic and inhomogeneous media.
{"title":"Toward the theory of semi-linear Beltrami equations","authors":"V. Gutlyanski̇i̇, O. Nesmelova, V. Ryazanov, E. Yakubov","doi":"10.33205/cma.1248692","DOIUrl":"https://doi.org/10.33205/cma.1248692","url":null,"abstract":"We study the semi-linear Beltrami equation $omega_{bar{z}}-mu(z) omega_z=sigma(z)q(omega(z))$ and show that it is closely related to the corresponding semi-linear equation of the form ${rm div} A(z)nabla,U(z)=G(z) Q(U(z)).$ Applying the theory of completely continuous operators by Ahlfors-Bers and Leray-Schauder, we prove existence of regular solutions both to the semi-linear Beltrami equation and to the given above semi-linear equation in the divergent form, see Theorems 1.1 and 5.2. We also derive their representation through solutions of the semi-linear Vekua type equations and generalized analytic functions with sources. Finally, we apply Theorem 5.2 for several model equations describing physical phenomena in anisotropic and inhomogeneous media.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48087745","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}
Bessel multipliers are operators defined from two Bessel sequences of elements of a Hilbert space and a complex sequence, and have frame multipliers as particular cases. In this paper an estimate of the spectral radius of a Bessel multiplier is provided involving the cross Gram operator of the two sequences. As an upshot, it is possible to individuate some regions of the complex plane where the spectrum of a multiplier of dual frames is contained.
{"title":"Estimate of the spectral radii of Bessel multipliers and consequences","authors":"R. Corso","doi":"10.33205/cma.1323956","DOIUrl":"https://doi.org/10.33205/cma.1323956","url":null,"abstract":"Bessel multipliers are operators defined from two Bessel sequences of elements of a Hilbert space and a complex sequence, and have frame multipliers as particular cases. In this paper an estimate of the spectral radius of a Bessel multiplier is provided involving the cross Gram operator of the two sequences. As an upshot, it is possible to individuate some regions of the complex plane where the spectrum of a multiplier of dual frames is contained.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42170073","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 $Omega$ be a bounded domain in $mathbb{R}^{n}$ with $C^{0,1}$ boundary, and let $d_{Omega}:Omegarightarrowmathbb{R}$ be the distance function $d_{Omega}left( xright) :=distleft( x,partialOmegaright) .$ Our aim in this paper is to study the existence and properties of principal eigenvalues of self-adjoint elliptic operators with weight function and singular potential, whose model problem is $-Delta u+bu=lambda mu$ in $Omega,$ $u=0$ on $partialOmega,$ $u>0$ in $Omega,$ where $b:Omega rightarrowmathbb{R}$ is a nonnegative function such that $d_{Omega}^{2}bin L^{infty}left( Omegaright) ,$ $m:Omegarightarrowmathbb{R}$ is a nonidentically zero function in $L^{infty}left( Omegaright) $ that may change sign, and the solutions are understood in weak sense.
{"title":"Principal eigenvalues of elliptic problems with singular potential and bounded weight function","authors":"T. Godoy","doi":"10.33205/cma.1272110","DOIUrl":"https://doi.org/10.33205/cma.1272110","url":null,"abstract":"Let $Omega$ be a bounded domain in $mathbb{R}^{n}$ with $C^{0,1}$ boundary, and let $d_{Omega}:Omegarightarrowmathbb{R}$ be the distance function $d_{Omega}left( xright) :=distleft( x,partialOmegaright) .$ Our aim in this paper is to study the existence and properties of principal eigenvalues of self-adjoint elliptic operators with weight function and singular potential, whose model problem is $-Delta u+bu=lambda mu$ in $Omega,$ $u=0$ on $partialOmega,$ $u>0$ in $Omega,$ where $b:Omega rightarrowmathbb{R}$ is a nonnegative function such that $d_{Omega}^{2}bin L^{infty}left( Omegaright) ,$ $m:Omegarightarrowmathbb{R}$ is a nonidentically zero function in $L^{infty}left( Omegaright) $ that may change sign, and the solutions are understood in weak sense.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46443705","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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