Hausdorff Matrices Research Articles

Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive Borel measure μ \mu on ( 0 , 1 ) (0,1) we consider an infinite matrix Γ μ \Gamma _\mu , related to the classical Hausdorff matrix defined by the same measure μ \mu , in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ \mu is the Lebesgue measure, Γ μ \Gamma _\mu reduces to the classical Hilbert matrix. We prove that the matrices Γ μ \Gamma _\mu are not Hankel, unless μ \mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H p , 1 ≤ p > ∞ H^p, \, 1 \leq p > \infty , and we study their compactness and complete continuity properties. In the case 2 ≤ p > ∞ 2\leq p>\infty , we are able to compute the exact value of the norm of the operator.

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem

We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval [a, b] and their associated second kind polynomials.

Degree OF Approximation of Functions f ( x , y ) by Double Hausdorff Matrix Summability Method

Degree OF Approximation of Functions f ( x , y ) by Double Hausdorff Matrix Summability Method

Upper bounds of some matrix operators on binomial and Orlicz-binomial double sequence spaces

In this article, we introduce binomial double sequence space bk(α,β;γ,δ)(1≤k≤∞) and Orlicz-binomial double sequence space bφ(α,β;γ,δ), and obtain certain inclusion results related to these spaces. We further focus on estimating the upper bounds for ∥M∥Lk,bk(α,β;γ,δ) and ∥M∥Lφ,bφ(α,β;γ,δ), where M is four dimensional Hausdorff matrix or four dimensional Nörlund matrix. We establish a Hardy type inequality in the case of four dimensional Hausdorff matrix.

On two resolvent matrices of the truncated Hausdorff matrix moment problem

We consider the truncated Hausdorff matrix moment problem (THMM) in case of a finite number of even moments to be called non degenerate if two block Hankel matrices constructed via the moments are both positive definite matrices. The set of solutions of the THMM problem in case of a finite number of even moments is given with the help of the block matrices of the so-called resolvent matrix. The resolvent matrix of the THMM problem in the non degenerate case for matrix moments of dimension $q\times q$, is a $2q\times 2q$ matrix polynomial constructed via the given moments. In 2001, in [Yu.M. Dyukarev, A.E. Choque Rivero, Power moment problem on compact intervals, Mat. Sb.-2001. -69(1-2). -P.175-187], the resolvent matrix $V^{(2n+1)}$ for the mentioned THMM problem was proposed for the first time. In 2006, in [A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval, Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. -2006. - 165. - P. 121-173], another resolvent matrix $U^{(2n+1)}$ for the same problem was given. In this paper, we prove that there is an explicit relation between these two resolvent matrices of the form $V^{(2n+1)}=A U^{(2n+1)}B$, where $A$ and $B$ are constant matrices. We also focus on the following difference: For the definition of the resolvent matrix $V^{(2n+1)}$, one requires an additional condition when compared with the resolvent matrix $U^{(2n+1)}$ which only requires that two block Hankel matrices be positive definite. In 2015, in [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259], a representation of the resolvent matrix of 2006 via matrix orthogonal polynomials was given. In this work, we do not relate the resolvent matrix $V^{(2n+1)}$ with the results of [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259]. The importance of the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$ is explained by the fact that new relations among orthogonal matrix polynomials, Blaschke-Potapov factors, Dyukarev-Stieltjes parameters, and matrix continued fraction can be found. Although in the present work algebraic identities are used, to prove the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$, the analytic justification of both resolvent matrices relies on the V.P. Potapov method. This approach was successfully developed in a number of works concerning interpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.

The accuracy and reliability of 3D printed aortic templates: a comprehensive three-dimensional analysis.

Advances in 3D printing technology allow us to continually find new medical applications. One of them is 3D printing of aortic templates to guide vascular surgeons or interventional radiologists to create fenestrations in the stent-graft surface for the implantation procedure called fenestrated endovascular aortic aneurysm repair. It is believed that the use of 3D printing significantly improves the quality of modified fenestrated stent-grafts. However, the accuracy and reliability of personalized 3D printed models of aortic templates are not well established. Thirteen 3D printed templates of the visceral aorta and sixteen of the aortic arch and their corresponding computer tomography of angiography images were included in this accuracy study. The 3D models were scanned in the same conditions on computed tomography (CT) and evaluated by three physicians experienced in vascular CT assessment. Model and patient CT measurements were performed at key landmarks to maintain quality for stent-graft modification, including side branches and aortic diameters. CT-scanned aortic templates were segmented, aligned with sourced patient data, and evaluated for the Hausdorff matrix. Next, Bland-Altman plots determined the degree of agreement. The Intraclass Correlation Coefficients values were more than 0.9 for all measurements of aortic diameters and aortic branches diameter in all landmark locations. Therefore, the reliability of the aortic templates was considered excellent. The Bland-Altman plots analysis indicated measurement biases of 0.05 to 0.47 for aortic arch templates and 0.06 to 0.38 for reno-visceral aortic templates. The arithmetic mean of Hausdorff's mean distances of the aortic arch templates was 0.47 mm (SD =0.06) and ranged from 0.34 to 0.58. The mean metrics for abdominal models was 0.24 mm (SD =0.03) and ranged from 0.21 to 0.31. The printed models of 3D aortic templates are accurate and reliable, thus can be widely used in endovascular surgery and interventional radiology departments as aortic templates to guide the physician-modified fenestrated stent-graft fabrication.

Characterizations of double Hausdorff matrices and best approximation of conjugate of a function in generalized Hölder space

In the present paper, we obtain characterization results of double Hausdorff matrices and apply these results to obtain error estimation of the function f?(x, y), conjugate to a function f (x, y) (periodic with period 2? in both x and y) in generalized H?lder space by double Hausdorff means of its double conjugate Fourier series. Some important corollaries are also obtained from our main result.

Domain of Padovan q-difference matrix in sequence spaces lp and l∞

In this study, we construct the difference sequence spaces lp (P?2q) = (lp)P?2q, 1 ? p ? ?, where P = (?rs) is an infinite matrix of Padovan numbers %s defined by ?rs = {?s/?r+5-2 0 ? s ? r, 0 s > r. and ?2q is a q-difference operator of second order. We obtain some inclusion relations, topological properties, Schauder basis and ?-, ?- and ?-duals of the newly defined space. We characterize certain matrix classes from the space lp (P?2q) to any one of the space l1, c0, c or l?. We examine some geometric properties and give certain estimation for von-Neumann Jordan constant and James constant of the space lp(P). Finally, we estimate upper bound for Hausdorff matrix as a mapping from lp to lp(P).

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Trigonometric approximation of functions f(x,y) of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y) belonging to the generalized Lipschitz class Lip ((xi _{1}, xi _{2} );r ), r geq 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip ((alpha ,beta );r ) and Lip(alpha ,beta ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C, gamma , delta ) means.

Approximation of functions in generalized Zygmund class by double Hausdorff matrix

In the present work, we emphasize, for the first time, the error estimation of a two-variable function g(y,z) in the generalized Zygmund class Y_{r}^{(xi )} (rgeq 1) using the double Hausdorff matrix means of its double Fourier series. In fact, in this work, we establish two theorems on error estimation of a two-variable function of g in the generalized Zygmund class.

A note on cosine series with coefficients of generalized bounded variation

Abstract In this note, we obtain a Tauberian theorem for a class of regular lower triangular matrices operating on cosine series with coefficients tending to zero. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

Three-term recurrence relation coefficients and continued fractions related to orthogonal matrix polynomials on the finite interval [a, b

ABSTRACT The four families of matrix orthogonal polynomials are considered arising in the truncated Hausdorff matrix moment (THMM) problem. Two of those families are associated with an odd number of moments and the other two with an even number of moments. The three-term recurrence relations associated with these four families are investigated. Certain explicit formulas are presented relating the three-term recurrence relation coefficients to the Dyukarev-Stieltjes parameters, the Schur complements and the orthogonal matrix polynomials associated with the THMM problem. The matrix version of the J-fraction is presented for the corresponding four extremal solutions of the THMM problem.

Estimation of upper bounds of certain matrix operators on Binomial weighted sequence spaces

In this article, we introduce binomial weighted sequence spaces $$b_p^{r,s}(w)$$ $$(1\le p<\infty )$$ , where $$w=(w_n)$$ is a non-negative decreasing sequence of real numbers, and investigate some topological and inclusion properties of the new spaces. We give an upper estimation of $$\left\| A\right\| _{\ell _p(w),b_p^{r,s}(w)}$$ , where A is the Hausdorff matrix operator or Norlund matrix operator. A Hardy type formula is established in the case of Hausdorff matrix operator. In the final section, we give an upper estimation for the transpose of Norlund matrix as an operator from $$\ell _p$$ to $$b_p^{r,s}$$ .

Some characterizations of Hausdorff matrices and their application to Fourier approximation

In this paper we characterize the Hausdorff matrices based on their mass functions. Using these characterizations we obtain the degree of approximation of functions belonging to weighted Lipschitz classes.

Lower bound of four-dimensional Hausdorff matrices

Let mathsf{H}=(h_{nmjk}) be a non-negative four-dimensional matrix. Denote by L_{p}(mathsf{H}) the supremum of those ℓ satisfying the following inequality: (∑n=0∞∑m=0∞(∑j=0∞∑k=0∞hnmjkxj,k)p)1/p≥ℓ(∑j=0∞∑k=0∞xj,kp)1/p,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ { \\Biggl( {\\sum_{n = 0}^{\\infty }{\\sum _{m = 0}^{\\infty } {{{ \\Biggl( {\\sum _{j = 0}^{\\infty }{\\sum_{k = 0}^{\\infty }{{h_{nmjk}} {x_{j,k}}} } } \\Biggr)}^{p}}} } } \\Biggr)^{1/p}} \\ge \\ell { \\Biggl( {\\sum_{j = 0}^{\\infty }{\\sum _{k = 0}^{ \\infty }{x_{j,k}^{p}} } } \\Biggr)^{1/p}}, $$\\end{document} where x=(x_{j,k}) in mathcal{L}_{p} with x_{j,k}ge 0. In this paper a Hardy type formula is established for L_{p}(mathsf{H}_{ mu times lambda }^{t}), where 0< ple 1 and mathsf{H}_{mu times lambda } is a four-dimensional Hausdorff matrix. A similar result is also obtained for the case in which mathsf{H}_{mu times lambda } is replaced by mathsf{H}_{mu times lambda }^{t}. As a consequence, we apply the results to some special four-dimensional Hausdorff matrices such as Cesàro, Euler, Hölder and Gamma matrices. Our results contain some generalizations of Copson’s discrete inequality.

Boundedness Problem of Four-Dimensional Matrices on the Domain Spaces of $$\mathcal {L}_p$$ L p

In this paper, we introduce and study the domain of an arbitrary four-dimensional summability matrix in the double sequence space $$\mathcal {L}_p$$ and focus on the boundedness problem of four-dimensional operators on it. We consider the class of four-dimensional Hausdorff matrices as operators mapping $$\mathcal {L}_p$$ into these domains and provide a Hardy type formula for their operator norms and lower bounds. Then we apply our results to some special domains of $$\mathcal {L}_p$$ such as the double Laurent, Taylor and Euler sequence spaces. Finally, we provide a general upper estimate for the operator norms of four-dimensional matrices which have real and complex entries with certain conditions.

Complementary Results on the Boundedness Problem of Factorizable Four-Dimensional Matrices

In this paper, the boundedness problem of factorizable four-dimensional matrices on the space of double sequences is investigated. As an application of our results, the lower bounds and the operator norms of four-dimensional Cesaro matrix and four-dimensional Copson matrix are obtained, which provide an extension of Hardy’s discrete inequality and Copson’s discrete inequality to double series, respectively. Finally, we present complementary results for the operator norm and lower bound of the four-dimensional Hausdorff matrices.

Norm inequalities involving upper bounds for certain matrix operators in Orlicz-type sequence spaces

In this paper, upper bounds of certain matrix operator norms are estimated in Orlicz-type weighted sequence spaces. Three spaces, namely, weighted Orlicz–Euler $$e_{\lambda , \varphi }^{\alpha }$$ , weighted Orlicz–Fibonacci $$F_{\lambda , \varphi }$$ and weighted Orlicz $$l_\varphi (\lambda )$$ are considered. Denote $$\Vert A\Vert _{X, Y}$$ as the operator norm of the matrix $$A=(a_{n, k})_{n, k\ge 0}$$ which maps X into Y, where X and Y are two normed sequence spaces. Then the evaluation of upper bounds for $$\Vert A\Vert _{l_\varphi (\lambda ), e_{\lambda , \varphi }^{\alpha }}$$ , $$\Vert A\Vert _{l_\varphi (\lambda ),F_{\lambda , \varphi }}$$ and $$\Vert A\Vert _{l_\varphi (\lambda ), l_\varphi (\mu )}$$ , where A is either Hausdorff or Norlund matrices is carried out throughout this paper. Some Hardy type formulas are established in case of Hausdorff matrices. Certain inclusion results are also obtained for each of the three sequence spaces. The results obtained in this work strengthen the results recently presented by Lashkaripour and Foroutannia (Proc Indian Acad Sci (Math Sci) 116(3):325–336, 2006) and Talebi and Dehghan (Linear Multilinear Algebra 62(10):1275–1284, 2014; Linear Multilinear Algebra 64(2):196–207, 2016).