Khizra Bukhsh, A. Younus, A. Mukheimer, T. Abdeljawad
The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.
{"title":"Fractional Proportional Linear Control Systems: A Geometric Perspective on Controllability and Observability","authors":"Khizra Bukhsh, A. Younus, A. Mukheimer, T. Abdeljawad","doi":"10.33205/cma.1454113","DOIUrl":"https://doi.org/10.33205/cma.1454113","url":null,"abstract":"The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141388234","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}
There are different methods available in literature to construct a new operator. One of the methods to construct an operator is the composition method. It is known that Baskakov operators can be achieved by composition of Post Widder $P_n$ and Sz'asz-Mirakjan $S_n$ operators in that order, which is a discretely defined operator. But when we consider different order composition namely $S_ncirc P_n$, we get another different operator. Here we study such and we establish some convergence estimates for the composition operators $S_ncirc P_n$, along with difference with other operators. Finally we found the difference between two compositions by considering numeric values.
{"title":"Convergence estimates for some composition operators","authors":"Vijay Gupta","doi":"10.33205/cma.1474535","DOIUrl":"https://doi.org/10.33205/cma.1474535","url":null,"abstract":"There are different methods available in literature to construct a new operator. One of the methods to construct an operator is the composition method. It is known that Baskakov operators can be achieved by composition of Post Widder $P_n$ and Sz'asz-Mirakjan $S_n$ operators in that order, which is a discretely defined operator. But when we consider different order composition namely $S_ncirc P_n$, we get another different operator. Here we study such and we establish some convergence estimates for the composition operators $S_ncirc P_n$, along with difference with other operators. Finally we found the difference between two compositions by considering numeric values.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141274255","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}
Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.
{"title":"Elementary proof of Funahashi's theorem","authors":"Yoshihro Sawano","doi":"10.33205/cma.1466429","DOIUrl":"https://doi.org/10.33205/cma.1466429","url":null,"abstract":"Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141040740","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 employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H"{o}lder type inequalities. Among other results, it is shown that if $mathbf{A}=(A_t)_{tin Omega}$ and $mathbf{B}=(B_t)_{tin Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${mathscr A}$ such that $int_{Omega}A_t,dmu(t)leq I_{mathscr A}$ and $int_{Omega}B_t,dmu(t)leq I_{mathscr A}$, and if $omega_f$ is an arbitrary operator mean with the representing function $f$, then begin{align*} left(I_{mathscr A}-int_{Omega}(A_t omega_f B_t),dmu(t)right)^p geqleft(I_{mathscr A}-int_{Omega}A_t,dmu(t)right) omega_{f^p}left(I_{mathscr A}-int_{Omega}B_t,dmu(t)right) end{align*} for all $0 < p leq 1$, which is an extension of the operator Bellman inequality.
{"title":"Extensions of the operator Bellman and operator Holder type inequalities","authors":"M. Bakherad, F. Kittaneh","doi":"10.33205/cma.1435944","DOIUrl":"https://doi.org/10.33205/cma.1435944","url":null,"abstract":"In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\"{o}lder type inequalities. Among other results, it is shown that if $mathbf{A}=(A_t)_{tin Omega}$ and $mathbf{B}=(B_t)_{tin Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${mathscr A}$ such that $int_{Omega}A_t,dmu(t)leq I_{mathscr A}$ and $int_{Omega}B_t,dmu(t)leq I_{mathscr A}$, and if $omega_f$ is an arbitrary operator mean with the representing function $f$, then\u0000 begin{align*}\u0000 left(I_{mathscr A}-int_{Omega}(A_t omega_f B_t),dmu(t)right)^p\u0000 geqleft(I_{mathscr A}-int_{Omega}A_t,dmu(t)right) omega_{f^p}left(I_{mathscr A}-int_{Omega}B_t,dmu(t)right)\u0000 end{align*}\u0000 for all $0 < p leq 1$, which is an extension of the operator Bellman inequality.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140262296","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 repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $bold R^n$ is represented by a hyperelliptic integral.
{"title":"On some general integral formulae","authors":"N. Ortner, P. Wagner","doi":"10.33205/cma.1406998","DOIUrl":"https://doi.org/10.33205/cma.1406998","url":null,"abstract":"We repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $bold R^n$ is represented by a hyperelliptic integral.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139878832","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 repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $bold R^n$ is represented by a hyperelliptic integral.
{"title":"On some general integral formulae","authors":"N. Ortner, P. Wagner","doi":"10.33205/cma.1406998","DOIUrl":"https://doi.org/10.33205/cma.1406998","url":null,"abstract":"We repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $bold R^n$ is represented by a hyperelliptic integral.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139819077","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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