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We consider infinite-dimensional generalized Hilbert matrices of the form H i , j = d i d j x i + x j , where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, for monotonically rearrangeable xi, also necessary conditions for di, xi such that the induced operator from l 2 → l 2 and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed. [ABSTRACT FROM AUTHOR]

We present two families of L-matrices and investigate their properties using intricate factorization techniques. These factorization schemes involve well-known matrices such as the classical Cesàro, Gamma, Hausdorff, and Hilbert matrices, which are utilized to assess the norm of L-matrices. Additionally, these factorizations give rise to novel and intriguing inequalities in sequence spaces. [ABSTRACT FROM AUTHOR]

It is well known that the Hilbert matrix H is bounded from the logarithmically weighted Bergman space A log α 2 into Bergman space A 2 when α > 2 . In this paper, we calculate lower bound and upper bound for the norm of the Hilbert matrix operator H from the logarithmically weighted Bergman space A log α 2 into Bergman space A 2 when α > 2 . We also calculate lower bound and upper bound for the norm of the Hilbert matrix operator from A log α p into A p , for 2 < p < ∞ and α > 1 . [ABSTRACT FROM AUTHOR]

In this study, by the composition of Cesàro operator with its transpose, we obtain two interesting operators which are Hilbert’s commutants. Moreover, the ℓ p norm of those operators has been computed in this study which results in obtaining some new inequalities, one of them is the generalization of Hardy’s inequality. As an application, we have also computed the norm of well-known operators on the matrix domains associated with those matrices. [ABSTRACT FROM AUTHOR]

In this paper, by using the triangle Hilbert matrix 𝐻 and the notion of ideal convergence for the sequences in intuitionistic fuzzy normed spaces, we introduce some new intuitionistic fuzzy normed sequence spaces as a domain of Hilbert matrix 𝐻, that is, 𝑐❑0(𝜇,𝜈) 𝐼 (𝐻) and 𝑐❑(𝜇,𝜈) 𝐼 (𝐻) Here, 𝑐❑0(𝜇,𝜈) 𝐼 (𝐻) denotes the Hilbert ideal null convergent sequence space with respect to the intuitionistic fuzzy norm and 𝑐❑(𝜇,𝜈) 𝐼 (𝐻) denotes the Hilbert ideal convergent sequence space with respect to the intuitionistic fuzzy norm We also define an open ball with respect to defined sequence space and prove that these open balls are the open sets of these spaces Further, we study some of its topological and algebraic properties We prove that these sequence spaces are linear spaces of In addition, we define a topology with respect to these sequence spaces and obtain that the defined topology is first countable and these topological sequence spaces are Hausdorff spaces We also obtain if and only if results that give an idea about when a sequence belonging to these spaces is classical convergent with respect to the intuitionistic fuzzy norm and when a sequence belonging to these spaces is ideal convergent with respect to the intuitionistic fuzzy norm [ABSTRACT FROM AUTHOR]

In this study, we obtain the ℓ p -norms of six classes of operators that commute with the infinite Hilbert operators. [ABSTRACT FROM AUTHOR]

We establish the mapping properties of the Bergman projections, the harmonic Bergman projections, the Cesáro operator and the Hilbert matrix on exponential Orlicz spaces and Zygmund spaces. We also have the mapping properties of the Berezin transform on Zygmund spaces. A sharpened result for the mapping properties of the Cauchy transform from the Lebesgue spaces to space of bounded function is also obtained. [ABSTRACT FROM AUTHOR]

In this research, we introduce some factorization for Hilbert operators of order n based on two important classes of Hausdorff operators, Cesàro and gamma. These factorizations lead us to some new inequalities, one is a generalized version of Hilbert's inequality. Moreover, as an application of our factorization, we compute the norm of Hilbert operators on some matrix domains. [ABSTRACT FROM AUTHOR]

In this article, we study some sequence spaces originated with an infinite Hilbert matrix and a Musielak-Orlicz function. Some topological and algebraic properties of new formed spaces are discuss. The α-, β- and β-duals are also determine. Finally, an attempt is made to characterize some matrix transformations between these spaces. Hilbert matrix is used for authentication, confidentiality to study cryptographic methods and frequently used in the security systems also. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of the domains ℓ p (Γ n) , c 0 (Γ n) , c (Γ n) and ℓ ∞ (Γ n) of the Gamma matrix of order n in the classical spaces ℓ p , c0, c and ℓ ∞ of absolutely p-summable, null, convergent and bounded sequences, respectively, and compute the α-, β- and γ-duals of these spaces. We characterize the classes of infinite matrices from the space ℓ p (Γ n) to the spaces ℓ ∞ and f, and from a normed sequence space to the gamma sequence spaces ℓ p (Γ n) , c 0 (Γ n) , c (Γ n) and ℓ ∞ (Γ n). Moreover, we introduce the necessary and sufficient conditions for factorizing an operator based on the weighted mean matrices and derive the factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Finally, we emphasize on the lower bound of operators from ℓ p into ℓ p (Γ n) , from ℓ p (Γ n) into ℓ p , from ℓ p (Γ n) into itself and from ℓ p into itself. [ABSTRACT FROM AUTHOR]

In this note, we obtain a new upper bound for the norm of the Hilbert matrix H on the weighted Bergman spaces A α p when - 1 < α < 0 , which is better than the known one in the case - 1 / 2 < α < 0 . [ABSTRACT FROM AUTHOR]

In this paper, by using the triangle Hilbert matrix H and the notion of ideal convergence for the sequences in intuitionistic fuzzy normed spaces, we introduce some new intuitionistic fuzzy normed sequence spaces as a domain of Hilbert matrix H, that is, c...0(μ,v) I (H) and c...(μ,v) I (H) Here, c...0(μ,v) I (H) denotes the Hilbert ideal null convergent sequence space with respect to the intuitionistic fuzzy norm and c...(μ,v) I (H) denotes the Hilbert ideal convergent sequence space with respect to the intuitionistic fuzzy norm We also define an open ball with respect to defined sequence space and prove that these open balls are the open sets of these spaces Further, we study some of its topological and algebraic properties We prove that these sequence spaces are linear spaces of In addition, we define a topology with respect to these sequence spaces and obtain that the defined topology is first countable and these topological sequence spaces are Hausdorff spaces We also obtain if and only if results that give an idea about when a sequence belonging to these spaces is classical convergent with respect to the intuitionistic fuzzy norm and when a sequence belonging to these spaces is ideal convergent with respect to the intuitionistic fuzzy norm [ABSTRACT FROM AUTHOR]

Let F denote the factorable matrix and X ∈ {ℓp, c0, c, ℓ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓp(F), ℓ∞), (ℓp(F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix. [ABSTRACT FROM AUTHOR]

In control theory and approximation theory, there naturally arise matrices which are the inverses of the Gram matrices for the monomial basis in the space of square integrable functions with respect to a measure. For example, such a matrix arises in the problem of finite time feedback stabilization of a linear system, and in the Hilbert problem on the minimal -norm of an integral polynomial. We show in a series of examples that the above inverse matrix is integral and has a large divisor. Our method is based on the arithmetic study of orthogonal polynomials naturally associated with the problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two factorizations for the Cesàro matrix of order n based on Cesàro and gamma matrices. The results of these factorizations are new inequalities, one of which is a generalized version of the well-known Hardy's inequality. Moreover, we obtain the norm of Cesàro operator of order n on Cesàro and gamma matrix domains. [ABSTRACT FROM AUTHOR]

We establish negative moment bounds for the minimum eigenvalue of the normalized Fisher information matrix in a stochastic regression model with a deterministic time trend. This result enables us to develop an asymptotic expression for the mean squared prediction error (MSPE) of the least squares predictor of the aforementioned model. Our asymptotic expression not only helps better understand how the MSPE is a affected by the deterministic and random components, but also inspires an intriguing proof of the formula for the sum of the elements in the inverse of the Cauchy/Hilbert matrix from a prediction perspective. [ABSTRACT FROM AUTHOR]

It is well known that the Hilbert matrix H is bounded on weighted Bergman spaces A α p if and only if 1 < α + 2 < p with the conjectured norm π / sin (α + 2) π p . The conjecture was confirmed in the case when α = 0 and also in the case when α > 0 and p ≥ 2 (α + 2) , which reduces the conjecture in the case when α > 0 to the interval α + 2 < p < 2 (α + 2) . In the remaining case when - 1 < α < 0 and p > α + 2 there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the Hilbert matrix H on weighted Bergman spaces A α p . In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when α > 0 and α + 2 < p < 2 (α + 2) . On the other hand, we also provide for the first time an explicit upper bound for the norm of the Hilbert matrix H on weighted Bergman spaces A α p in the case when - 1 < α < 0 and p > α + 2 . [ABSTRACT FROM AUTHOR]

Image encryption based on elliptic curve cryptosystem and reducing its complexity is still being actively researched. Generating matrix for encryption algorithm secret key together with Hilbert matrix will be involved in this study. For a first case we will need not to compute the inverse matrix for the decryption processing cause the matrix that be generated in encryption step was self invertible matrix. While for the second case, computing the inverse matrix will be required. Peak signal to noise ratio (PSNR), and unified average changing intensity (UACI) will be used to assess which case is more efficiency to encryption the grayscale image. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of the domains c0(Cn), c(Cn), and ℓp(Cn) with 0 < p < 1 of the Cesàro matrix of order n in the classical spaces c0, c, and ℓp of null, convergent, and absolutely p‐summable sequences, respectively, and compute the α‐, β‐, and γ‐duals of these spaces. We characterize the classes of infinite matrices from the space ℓp(Cn) to the spaces ℓ∞, c, and c0 and from a normed sequence spaces to the sequence spaces c0(Cn), c(Cn), and ℓp(Cn). Moreover, we compute the lower bound of operators from ℓp into ℓp(Cn), from ℓp(Cn) into ℓp and from ℓp(Cn) into itself. [ABSTRACT FROM AUTHOR]

The problem of constructing examples that establish the sharpness of the Menchoff– Rademacher theorem on the Weyl multiplier for the almost everywhere convergence of series in general orthogonal systems is considered. We construct an example of a discrete orthonormal system based on blocks such that the partial sums of the series in this system has majorants whose -norm increases as . This orthonormal system is generated by an orthogonal matrix that has improved characteristics, as compared to the Hilbert matrix. We continue the study of B. S. Kashin, who constructed an example: of an orthonormal system, based on binary blocks, with majorant of partial sums increasing as . [ABSTRACT FROM AUTHOR]

Let Δ − n be the backward difference operator of order −n. In this paper, we study some properties of the matrix domain associated with this operator and after computing α-, β-, γ-duals, we characterize some matrix classes on this space. Moreover, we investigate the problem of finding the norm of well-known operators such as Cesàro and Hilbert from ℓ p (Δ − n) into ℓ p. [ABSTRACT FROM AUTHOR]

In the present study, we construct a new matrix which we call quasi-Cesàro matrix and is a generalization of the ordinary Cesàro matrix, and introduce BK-spaces Ckq and C∞q as the domain of the quasi-Cesàro matrix Cq in the spaces ℓk and ℓ∞, respectively. Furthermore, we exhibit some topological properties and inclusion relations related to these newly defined spaces. We determine the basis of the space Ckq and obtain Köthe duals of the spaces Ckq and C∞q. Based on the newly defined matrix, we present a factorization for the Hilbert matrix and generalize Hardy's inequality, as an application. Moreover we find the norm of this new matrix as an operator on several matrix domains. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of the domain of the Cesàro matrix of order n and compute Köthe dual of this space. Moreover, we compute the norm of well-known operators such as Hilbert, Hausdorff, backward difference and weighted mean operators on this new sequence space. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the problem of finding the norm of Hilbert operator on some sequence spaces. Meanwhile, we obtain several interesting inequalities and inclusions. [ABSTRACT FROM AUTHOR]

In this research, we investigate two types of Copson matrices, the generalized Copson matrix and the Copson matrix of order n, and their associated sequence spaces generated by these matrices. We also investigate the topological properties, inclusions, and dual spaces of these new Banach spaces as well as compute the norm of Copson operators on the well-known matrix domains such as Hilbert and difference sequence spaces. Moreover, in a reverse manner, we investigate the norm of well-known operators on the Copson matrix domains generated with Copson matrices. Through this study we introduce several new inequalities, inclusions, and factorizations for well-known operators. [ABSTRACT FROM AUTHOR]

In this paper, two new analogues of the Hilbert matrix with four-parameters have been introduced. Explicit formulæ are derived for the LU-decompositions and their inverses, and the inverse matrices of these analogue matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce factorizations for the Hilbert and Cesàro matrices based on generalized Cesàro matrix and as the application, we generalize the Hardy's inequality versus Hilbert's. [ABSTRACT FROM AUTHOR]

It is known that if X and Y are spaces of holomorphic functions in the unit disc D , which are between the mean Lipschitz space Λ 1 / p p , where 1 < p < ∞ , and the Bloch space B , then the generalized Hilbert matrix H μ , induced by a positive Borel measure μ on the interval [0, 1), is a bounded operator from the space X into the space Y if and only if μ is a 1-logarithmic 1-Carleson measure. We improve this result by proving that the same conclusion holds if we replace the space Λ 1 / p p , 1 < p < ∞ , by the space Λ 1 1 . Also we prove that the same conclusion holds if X and Y are spaces of holomorphic functions in D , which are between the Besov space B 1 , 1 and the mixed norm space H ∞ , 1 , 1 . As immediate consequences, we obtain many results and some of them are new. [ABSTRACT FROM AUTHOR]

Very recently, Božin and Karapetrovi'c [J. Funct. Anal. 274 (2018), no. 2, pp. 525-543] solved a conjecture by proving that the norm of the Hilbert matrix operator H on the Bergman space Ap is equal to π/sin(2π p) for 2 < p < 4. In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of H defined on the Korenblum spaces H∞α for 0 < α ≤ 2/3 and an upper bound for the norm on the scale 2/3 < α < 1. [ABSTRACT FROM AUTHOR]

Let be a non-negative matrix and be the forward difference operator of order n for . Denote by the infimum of those U, satisfying the following inequality: where and . This paper is focused on the problem of finding , where A is a matrix operator from into . In particular, we apply our results to well known matrix operators such as Copson, Hilbert, and Nörlund. [ABSTRACT FROM AUTHOR]

In this paper, some applications of the Hilbert matrix in image processing and cryptology are mentioned and an algorithm related to the Hilbert view of a digital image is given. New matrix domains are constructed and some of their properties are investigated. Furthermore, dual spaces of new matrix domains are computed and matrix transformations are characterized. Finally, examples of transformations of new spaces are given. [ABSTRACT FROM AUTHOR]

Let be a matrix operator, and be two non-negative sequences and . This paper is focused on the problem of finding the infimum of those U, satisfying the following inequality: for all sequences . We apply our results to well-known matrix operators such as backward difference, quasi-summability, weighted mean, Hilbert and Nörlund. [ABSTRACT FROM AUTHOR]

We prove that the Markov–Stieltjes transform is a bounded non compact Hankel operator on Hardy spacewith Hilbert matrix with respect to the standard Schauder basis ofand a bounded non-compact operator on Lebesgue spaceforand obtain estimates for its norm in this spaces. It is shown that the Markov–Stieltjes transform onis unitary equivalent to the Markov–Stieltjes transform on. Inverse formulas and operational properties for this transform are obtained. [ABSTRACT FROM AUTHOR]

This article continues the study of the so-called direct discontinuous Galerkin (DDG) method for diffusion problems as developed in [Liu and Yan, SIAM J Numer Anal 47 (2009), 475-698;, Liu and Yan, Commun Comput Phys 8 (2010), 541-564; C. Vidden and J. Yan, J Comput Math 31 (2013), 638-662; H. Liu, Math Comp (in press)]. A key feature of the DDG method lies with the numerical flux design which includes two (or more) free parameters. This article identifies the class of all admissible numerical flux choices (Theorem 2.2) for degree n polynomial approximations for the symmetric DDG method [C. Vidden and J. Yan, J Comput Math 31 (2013), 638-662], guaranteeing stability of the resulting method. Our main contribution is the new technique of analysis for the DDG admissibility condition. The strategy is to directly evaluate the admissibility condition (Lemma 2.4) by choosing a simple polynomial basis. The admissibility condition is then transformed into an eigenvalue problem resulting in showing needed properties of inverse Hilbert matrices (Lemma 2.3, Appendix). Numerical tests are provided to confirm theoretical results. [ABSTRACT FROM AUTHOR]

An accurate numerical method is established for matrix inversion. It is shown theoretically that the scheme possesses the high order of convergence of seven. Subsequently, the method is taken into account for solving linear systems of equations. The accuracy of the contributed iterative method is clarified on solving numerical examples when the coefficient matrices are ill-conditioned. All of the computations are performed on a PC using several programs written in. Mathematica 7 [ABSTRACT FROM AUTHOR]

For each noninteger complex number λ, the Hilbert matrix defines a bounded linear operator on the Hardy spaces $\mathcal{H}^{p}$, 1< p<∞, and on the Korenblum spaces $\mathcal{A}^{-\tau}$, τ>0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex λ results by Hill and Rosenblum for real λ. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix. [ABSTRACT FROM AUTHOR]

We consider the action of the Hilbert matrix operator, H; on the Hardy space H1, weighted Hardy spaces Hαp (α ≥ 0); Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps Hαp into Hαp if and only if α+1/p < 1. A criterion for H f to belong to H1 is given provided the coefficients of f are nonnegative. Also, H maps the A2-space with weight logα(2/(1 - |z|2)) into the ordinary Bergman space A2 if α > 3. Similarly, the Bloch space with logarithmic weight is mapped by H into the ordinary Bloch space. [ABSTRACT FROM AUTHOR]

We consider the action of the Hilbert matrix operator, H, on the Hardy space H1, weighted Hardy spaces Hp &agr; (&agr; ≥ 0), Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps Hp &agr; into Hp &agr; if and only if &agr;+1=p < 1: A criterion for Hf to belong to H1 is given provided the coefficients of f are nonnegative. Also, H maps the A2-space with weight log&agr;(2=(1 - &verbar;z&verbar;2)) into the ordinary Bergman space A2 if &agr; > 3: Similarly, the Bloch space with logarithmic weight is mapped by H into the ordinary Bloch space. [ABSTRACT FROM AUTHOR]

In this research, we firstly introduce a criterion for factorizing an operator based on the gamma operator, and as a result we present two factorizations for the Cesàro and Hilbert operators. As another point of view, we study some properties of the matrix domain associated with the gamma matrix of order n and compute the duals of this space. Moreover, we compute the norm of well-known operators into, from, and on the gamma sequence space. [ABSTRACT FROM AUTHOR]

We characterize the Jordan domain of a given perimeter L > 4 such that the two designated interior points ±1 lying in its interior have the smallest hyperbolic distance between them. We also show how this result serves as an example illustrating the theory of quadrature domains studied by Björn Gustafsson and Harold Shapiro. [ABSTRACT FROM AUTHOR]

In this paper, we considered the problem of finding the upper bound Hausdorff matrix operator from sequence spaces lp(v) (or d(v, p)) into lp(w) (or d(w, p)). Also we considered the upper bound problem for matrix operators from d(v, 1) into d(w, 1), and matrix operators from e(w, ∞) into e(v, ∞), and deduce upper bound for Cesaro, Copson and Hilbert matrix operators, which are recently considered in [5] and [6] and similar to that in [10]. [ABSTRACT FROM AUTHOR]

We generalize the construction proposed by A. Paskiewich of an example of an orthonormal system which establishes the sharpness of the Men’shov-Rademacher theorem. The connection of his example with that of Men’shov is elucidated. [ABSTRACT FROM AUTHOR]

In the present paper, the author gives analytic expressions for the Cholesky decomposition of the Hilbert matrix H=[1/( j+k−1)]= LDL , and the inverse matrices. The results are given in the formulas (2)-(4). [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of the domains c (C n) , c 0 (C n) , and ℓ p (C n) (0 < p < 1) of the Copson matrix of order n, where c, c 0 , and ℓ p are the spaces of all convergent, convergent to zero, and p-summable real sequences, respectively. Moreover, we compute the Köthe duals of these spaces and the lower bound of well-known operators on these sequence spaces. The domain ℓ p (C n) of Copson matrix C n of order n in the sequence space ℓ p , the norm of operators on this space, and the norm of Copson operator on several matrix domains have been investigated recently in (Roopaei in J. Inequal. Appl. 2020:120, 2020), and the present study is a complement of our previous research. [ABSTRACT FROM AUTHOR]