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This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ K = K − 1 K = 1 . Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

Let Delta(-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 alpha-, beta-, gamma-duals, we characterize some matrix classes on this space. Moreover, we investigate the problem of finding the norm of well-known operators such as Cesaro and Hilbert from l(p)(Delta(-n)) into l(p). WOS:000628705400001 2-s2.0-85102678775

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 $$4\times 4$$ blocks such that the partial sums of the series in this system has majorants whose $$L_2$$ -norm increases as $$\log_2N$$ . 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 $$\sqrt{\log_2N}$$ .

Roopaei in [13] has introduced some factorization for the infinite Hilbert matrix and the Cesaro matrix of order n based on the generalized Cesaro matrix. In this research, we investigate the norm of these two operators on the generalized Cesaro matrix domain. Moreover we introduce some factorizations for the Hilbert matrix. Hence the present study is a complement of Roopaei’s research. There are several new Banach spaces who have introduced and studied by using matrix domains of special lower triangular matrices. For more references we encourage the readers to some papers [1, 3, 17, 18] and textbook [2].

It is well known that the Hilbert matrix $${\mathrm {H}}$$ is bounded on weighted Bergman spaces $$A^p_\alpha $$ if and only if $$10$$ and $$p\ge 2(\alpha +2)$$ , which reduces the conjecture in the case when $$\alpha >0$$ to the interval $$\alpha +2\alpha +2$$ 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 $${\mathrm {H}}$$ on weighted Bergman spaces $$A^p_\alpha $$ . In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when $$\alpha >0$$ and $$\alpha +2\alpha +2$$ .

In this paper, we study the boundedness of the matrix operators from weighted sequence spaces into Copson weighted sequence spaces and deduce several interesting inequalities. Moreover, we also make an effort to compute the norm of various matrix operators such as forward difference, transpose of Norlund, transpose of weighted mean and Hilbert matrix.

BackgroundCurrent intra-tumoral heterogeneous feature extraction in radiology is limited to the use of a single slice or the region of interest within a few context-associated slices, and the decoding of intra-tumoral spatial heterogeneity using whole tumor samples is rare. We aim to propose a mathematical model of space-filling curve-based spatial correspondence mapping to interpret intra-tumoral spatial locality and heterogeneity.MethodsA Hilbert curve-based approach was employed to decode and visualize intra-tumoral spatial heterogeneity by expanding the tumor volume to a two-dimensional (2D) matrix in voxels while preserving the spatial locality of the neighboring voxels. The proposed method was validated using three-dimensional (3D) volumes constructed from lung nodules from the LIDC-IDRI dataset, regular axial plane images, and 3D blocks.ResultsDimensionality reduction of the Hilbert volume with a single regular axial plane image showed a sparse and scattered pixel distribution on the corresponding 2D matrix. However, for 3D blocks and lung tumor inside the volume, the dimensionality reduction to the 2D matrix indicated regular and concentrated squares and rectangles. For classification into benign and malignant masses using lung nodules from the LIDC-IDRI dataset, the Inception-V4 indicated that the Hilbert matrix images improved accuracy (85.54% vs. 73.22%,p ConclusionsOur study indicates that Hilbert curve-based spatial correspondence mapping is promising for decoding intra-tumoral spatial heterogeneity of partial or whole tumor samples on radiological images. This spatial-locality-preserving approach for voxel expansion enables existing radiomics and convolution neural networks to filter structured and spatially correlated high-dimensional intra-tumoral heterogeneity.

Very recently, Bo\v{z}in and Karapetrovi\'c solved a conjecture by proving that the norm of the Hilbert matrix operator $\mathcal{H}$ on the Bergman space $A^p$ is equal to $\frac{\pi}{\sin(\frac{2\pi}{p})}$ 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 $\mathcal{H}$ defined on the Korenblum spaces $H^\infty_\alpha$ for $0 < \alpha \le 2/3$ and an upper bound for the norm on the scale $2/3 < \alpha < 1$., Comment: 10 pages

The Hilbert matrix $$\begin{aligned} {\mathcal {H}}_\lambda =\left( \frac{1}{n+m+\lambda }\right) _{n,m=0}^{\infty }, \quad \lambda \ne 0,-1,-2, \ldots \, \end{aligned}$$ H λ = 1 n + m + λ n , m = 0 ∞ , λ ≠ 0 , - 1 , - 2 , … generates a bounded linear operator in the Hardy spaces $$H^p$$ H p and in the $$l^p$$ l p -spaces. The aim of this paper is to study the spectrum of this operator in the spaces mentioned. In a sense, the presented investigation continues earlier works of various authors. More information concerning the history of the topic can be found in the introduction.

In this paper, we introduce Cesaro–Gamma matrix that exhibits the structure of both the Cesaro and Gamma matrices. We study the domain of this new matrix in the space $$\ell _p$$ ( $$1\le p\le \infty$$ ). By this new matrix, we obtain a factorization for the infinite Hilbert matrix, based on the Cesaro matrix of order $$\lambda$$ , of the form $$H=B^\lambda C^\lambda$$ . As a second application of this operator, we obtain a factorization for the Cesaro matrix of order $$\lambda$$ of the form $$C^{\lambda +\tilde{\lambda }}=R^{\lambda , \tilde{\lambda }+1}C^{\tilde{\lambda }}$$ , which results in a factorization for the Cesaro matrices of the form $$C^\lambda =S^{\lambda ,\tilde{\lambda }} C^{\tilde{\lambda }}.$$

The fractional Hilbert transform (FHT) is a linear combination of a signal and its Hilbert transform, and invites extensions of the classical analytic signal. Mathematical properties of the FHT and three possible versions of fractional analyticity are explored in detail.

Kilic, Emrah/0000-0003-0722-7382 WOS:000519766200005 In this paper, two new analogues of the Hilbert matrix with four-parameters have been introduced. Explicit formul ae are derived for the LU-decompositions and their inverses, and the inverse matrices of these analogue matrices. (C) 2020 Mathematical Institute Slovak Academy of Sciences

In this article, the open problem of finding the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces $A^p_\alpha$ is adressed. The norm was conjectured to be $\frac{\pi}{\sin \frac{(2+\alpha)\pi}{p}}$ by Karapetrovi\'{c}. We obtain a complete solution to the conjecture for $\alpha \ge 0$ and $2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3} \le p < 2(2+\alpha)$ and a partial solution for $2+2\alpha < p < 2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3}.$ Moreover, we also show that the conjecture is valid for small values of $\alpha$ when $2+2\alpha < p \le 3+2\alpha.$ Finally, the case $\alpha = 1$ is considered., Comment: 21 pages

In this paper, we introduce factorizations for the Hilbert and Cesaro matrices based on generalized Cesaro matrix and as the application, we generalize the Hardy's inequality versus Hilbert's.

We show that if α > 1, then the logarithmically weighted Bergman space $$A_{{{\log }^\alpha }}^2$$ is mapped by the Libera operator L into the space $$A_{{{\log }^{\alpha - 1}}}^2$$ , while if α > 2 and 0 < e ≤ α−2, then the Hilbert matrix operator H maps $$A_{{{\log }^\alpha }}^2$$ into $$A_{{{\log }^{\alpha - 2 - \varepsilon }}}^2$$ . We show that the Libera operator L maps the logarithmically weighted Bloch space $${B_{{{\log }^\alpha }}}$$ , α ∈ R, into itself, while H maps $${B_{{{\log }^\alpha }}}$$ into $${B_{{{\log }^{\alpha + 1}}}}$$ . In Pavlovic’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ . We show that this result is sharp. We also show that H maps $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ and that this result is sharp also.

Let A=(ak,m)k,m≥0 be a non-negative matrix and Δn be the forward difference operator of order n for n=1,2,⋯ . Denote by ‖A‖p,Δn the infimum of those U, satisfying the following inequality: ∑k=0∞∑j=0n∑m=0∞(-1)jnjak+j,mxmp1p≤U∑j=0∞|xj|p1p, where x∈lp and p≥1 . This paper is focused on the problem of finding ‖A‖p,Δn , where A is a matrix operator from lp into lp(Δn) . In particular, we apply our results to well known matrix operators such as Copson , Hilbert, and Norlund.

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...

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space A p into A p if and only if 2 p ∞ . In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space A p is equal to π sin ⁡ 2 π p , when 4 ≤ p ∞ , and it was also conjectured that ‖ H ‖ A p → A p = π sin ⁡ 2 π p , when 2 p 4 . In this paper we prove this conjecture.

It is well known (see [8] , [14] ) that the Libera operator L is bounded on the Besov space H ν p , q , α if and only if 0 κ p , α , ν : = ν − α − 1 p + 1 . We prove unexpected results: the Hilbert matrix operator H, as well as the modified Hilbert operator H ˜ , is bounded on H ν p , q , α if and only if 0 κ p , α , ν 1 . In particular, H, as well as H ˜ , is bounded on the Bergman space A p , α if and only if 1 α + 2 p and is bounded on the Dirichlet space D α p = A 1 p , α if and only if max ⁡ { − 1 , p − 2 } α 2 p − 2 . Our results are substantial improvement of [11, Theorem 3.1] and of [6, Theorem 5] .

Lim and Palfia established the notion of the matrix power means for k positive definite matrices ( k ≥ 3 ) and showed that the matrix power means have the information monotonicity for a unital positive linear mapping. In this note, by virtue of the generalized Kantorovich constant, we show counterparts to the information monotonicity of the matrix power means.

The famous Douglas theorem contains four equivalent conditions on the solvability of the operator equation A X = B associated to Hilbert spaces, which was generalized to the Hilbert C ⁎ -module case by Fang et al. (2009) [5] . In this paper we prove that in the Hilbert C ⁎ -module case, the majorization condition and the range inclusion condition are in fact not equivalent.

In this paper we obtain some results concerning the ascent and descent of a quasi-Fredholm relation in a Hilbert space and we analyze the behaviour of a polynomial in a quasi-Fredholm relation in a Hilbert space.

In this paper, we introduce a factorization for the infinite Hilbert matrix based on Cesaro matrices. Moreover, through the relation between Cesaro and Gamma matrices, we extract our second factorization for the Hilbert matrix based on Gamma matrices. The results of these factorizations are two new inequalities one of which is a generalized version of the well-known Hilbert’s inequality.

A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on l 2 ( N 0 ) , is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families.

In this note, we study the asymptotics of the determinant $\det(I_N - \beta H_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq 1$. Moreover, we assume $\beta\in\mathbb C$ with $|\beta

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $$N\times N$$N×N truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).

Remote satellite imaging provides vital information for observing number of applications such as, land region detection and urban area classification. This paper proposed a novel approach for Classification and extraction of texture features from high resolution satellite images dataset. Preprocessing is done for satellite sensing image using Hilbert matrix filter and Modified Hilbert matrix filter. Then the texture features are extracted from the Hilbert image and Modified Hilbert image using Gray Level Co-occurrence Matrix (GLCM). Finally, obtained features are classified using Decision tree and Random Forest and the accuracy, precision, recall and F-measure is analyzed for performance evaluation.

One of the important issues in the weighted least-squares (WLS) design of two-dimensional (2-D) finite impulse response (FIR) filters is the computational complexity of the design algorithms. This paper presents a matrix formulation of the design problem and derives a matrix-based generalized conjugate gradient (GCG) algorithm. By defining a linear operator acting on the coefficient matrix of the filter, the optimality condition of the design problem is expressed as a linear operator equation. By proving the boundedness, self-adjointness and positive-definiteness of the linear operator in the Hilbert space of the coefficient matrix, a GCG algorithm designated for the Hilbert matrix space is used to solve the linear operator equation. The convergence of the proposed algorithm in finite steps is established and its high computational efficiency is demonstrated by design examples and comparison with some existing methods.

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal complexity for this task, and prove approximation bounds. Numerical examples illustrate our theoretical results. (C) 2015 Elsevier Inc. All rights reserved.

In this paper we study propagation phenomena for mono-weakly hyponormal 2-variable weighted shifts. Next, we describe the positivity and determinant of an immediate ancestor of the classical Hilbert matrix. Finally, using its positivity, we construct a family of examples which shows and elucidates the gap between k-hyponormality and ( k + 1 ) -hyponormality for commuting pairs of operators for each k ≥ 1 .

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ {\mathcal H}_\mu = (\mu_{n+k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $ induces the operator $ {\mathcal H}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty\mu_{n,k}a_k\right)z^n $ on the space of all analytic functions $ f(z) = \sum^\infty_{n = 0}a_nz^n $ in the unit disk $ {\mathbb{D}} $. In this paper, we characterize the boundedness and compactness of $ {\mathcal H}_\mu $ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $ {\mathcal H}_\mu $ from $ {\mathcal{B}}^\alpha $ to $ {\mathcal{B}} $ and BMOA.

Several applications have problems with the representation of the real numbers because of its drawbacks like the propagation and the accumulation of errors. These numbers have a fixed length format representation that provides a large dynamic range, but on the other hand it causes truncation of some parts of the numbers in case of a number that needs to be represented by a long stream of bits. Researchers suggested many solutions for these errors, one of these solutions is the Multi-Number (MN) system. MN system represents the real number as a vector of floating-point numbers with controlled accuracy by adjusting the length of the vector to accumulate the non-overlapping real number sequences. MN system main drawback is the MN computations that are iterative and time consuming, making it unsuitable for real time applications. In this work, the Single Instruction Multiple Data (SIMD) model supported in modern CPUs is exploited to accelerate the MN Computations. The basic arithmetic operation algorithms had been adjusted to make use of the SIMD architecture and support both single and double precision operations. The new architecture maintains the same accuracy of the original one, when was implemented for both single and double precision. Also, in this paper the normal Gaussian Jordan Elimination algorithm was proposed and used to get the inverse of the Hilbert Matrix, as an example of ill-conditioned matrices, instead of using iterative and time-consuming methods. The accuracy of the operations was proved by getting the inverse of the Hilbert Matrix and verify that the multiplication of the inverse and the original matrix producing the unity matrix. Hilbert Matrix inverse execution time was accelerated and achieved a speedup 3x, compared to the original NM operations. In addition to the previous, the accelerated MN system version was used to solve the polynomial regression problem.

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. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,\pi]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem., Comment: 18 pages, to appear in Arkiv f\"or Matematik

A new viewpoint is used to understand the generation process of the Hilbert curve. A one-to-one correspondence between the 4-adic expansion of the unit interval and the fractal curve’s iterative generating process is established, and an analytical expression of the level-n Hilbert curve is obtained. This expression can take limit and represent the curve with 2-adic series. Although composition of functions this expression can substitute the generator of the Hilbert curve, while it can be proved, using the expression, that the generation of the Hilbert curve depends on how the subsquares are connected rather than the shape of the generator.

Let n be any nonnegative integer. Let V = Pn be the vector space of polynomials of degree at most n, equipped with the inner product ⟨f, g⟩ = ∫10f(x)g(x) dx. Let D: V → V be the differentiation operator, D(f) = f′. Then D has an adjoint D*. We have closed-form expressions for D*, which were conjectured by computing D* for small values of n and finding a pattern. (If f(x) is a polynomial of degree k ≤ n, then while the value of D(f(x)) is independent of n, the value of D*(f(x)) depends on n.) We also find formulas for D* in terms of classical Legendre polynomials, shifted to the interval [0, 1]. Using these formulas, it is easy to prove that our closed-form expressions are correct. An alternative approach yields combinatorial identities involving the entries of the inverses of Hilbert matrices.

The Hilbert transform is an important tool in image processing and optics. The Hilbert transform can be generalized to a fractional Hilbert transform. The generalization is driven by optics and image processing. We will generalize the fractional Hilbert transform into 2 dimensions by rotating the Hilbert transform in \({\mathbb{R}^{3}}\). The definition of the Hilbert transform as well as of the rotations will be done by quaternions.

In this paper, a practical bivariate Newton–Thiele type matrix Pade approximation is introduced by using the generalized inverse of a matrix. The approximants are expressed in the form of Newton expansion and branched continued fractions, which can be computed by an efficient recursive algorithm. Some algebraic properties are also discussed and the application in state-space realization of the 2-D filters is given in the end.

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

The Filbert matrix \(\mathcal {F}_{N}\) has entries \(\frac{1}{F_{i+j-1}}\) and is an analogue of the Hilbert matrix where \(F_{n}\) is the \(n\)th Fibonacci number. Various extensions are nowadays known with beautiful explicit results (LU-decomposition, Cholesky decompositions, etc.). The inverse Filbert matrix does, in these general instances, not have closed form entries. Nevertheless, it can be decomposed itself in very much the same way as the matrix. These results are not corollaries of the decomposition of the Filbert matrix itself.

In this paper, firstly, some applications of Hilbert matrix in image processing and cryptology are mentioned and an algorithm related to the Hilbert view of a digital image is given. In section 2, the new matrix domains are constructed and some properties are investigated. Furthermore, dual spaces of new matrix domains are computed and matrix transformations are characterized. In last section, examples of transformations of new spaces are given., 14 pages, 2 figures, 4 table. arXiv admin note: text overlap with arXiv:1602.04103

To each semidefinite program (SDP) in primal form, we associate the matrix algebra generated by its constraint matrices. In this note, we show that this algebra is always a full matrix algebra for SDPs arising from (commutative or non-commutative) sum of squares (SOS) problems. For SDPs arising from non-commutative SOS and commutators problems, the situation is less clear. We identify an exceptional case, where the corresponding matrix algebra is not the full matrix algebra, and use it to reprove the Burgdorf–Klep non-commutative variant of Hilbert’s ternary quartics theorem: a bivariate non-commutative polynomial of degree at most is trace positive if it is a sum of four squares and commutators.

The Hankel matrix has various applications. In this paper we prove that Hankel matrix is strongly regular and apply to obtain the necessary and sufficient conditions to sum the Walsh-Fourier series of a function of bounded variation.