Hankel Matrix Research Articles

Bidiagonal Factorizations of Filbert and Lilbert Matrices

Extensions of Filbert and Lilbert matrices are addressed in this work. They are reciprocal Hankel matrices based on Fibonacci and Lucas numbers, respectively, and both are related to Hilbert matrices. The Neville elimination is applied to provide explicit expressions for their bidiagonal factorization. As a byproduct, formulae for the determinants of these matrices are obtained. Finally, numerical experiments show that several algebraic problems involving these matrices can be solved with outstanding accuracy, in contrast with traditional approaches.

Deblending and interpolation of subsampled blended seismic data based on damped randomized singular spectrum analysis

AbstractWhen compared to traditional seismic data acquisition, irregular blended acquisition significantly promotes the acquisition efficiency. Yet, the blending noise of subsampled blended data introduces new obstacles for the subsequent processing of seismic data. Due to the predictability of linear events in the frequency–space domain, the constructed Hankel matrices exhibit low‐rank characteristics. However, the blending noise of subsampled blended data increases the rank, so deblending and interpolation can be implemented via rank‐reduction algorithms such as the singular spectrum analysis. The significant computing cost of the singular value decomposition, however, makes the traditional singular spectrum analysis inefficient. An alternative algorithm, known as the randomized singular spectrum analysis, employs the randomized singular value decomposition instead of the traditional singular value decomposition for rank‐reduction. The randomized singular spectrum analysis significantly enhances the efficiency of the decomposition process, particularly when dealing with large Hankel matrices. There still remains some random noise when using the singular spectrum analysis or randomized singular spectrum analysis for subsampled blended data, because the noise subspace and signal subspace are coupled together. Thus, we incorporate a damping operator into the randomized singular value decomposition and propose a novel damped randomized singular spectrum analysis method. The damped randomized singular spectrum analysis combines the advantages of the randomized singular value decomposition and the damping operator to enhance the computational efficiency and suppress the remaining noise. Moreover, an iterative projected gradient descent strategy is introduced to achieve deblended and interpolated seismic data for subsequent processing. Examples from synthetic data and field data are used to demonstrate the effectiveness and superiority of the proposed damped randomized singular spectrum analysis method, which enhances the accuracy and efficiency during simultaneous deblending and interpolation.

Commuting Toeplitz and small Hankel operators on the Bergman space

Commuting Toeplitz and small Hankel operators on the Bergman space

Isolation and identification of rolling bearing compound faults based on adaptive periodized singular spectrum analysis and Rényi entropy

Due to the coupling of multiple fault feature information and contamination of heavy background noise, it is a challenging task to accurately identify rolling bearing compound faults (RBCFs). A method for isolating and identifying the RBCF is proposed by integrating adaptive periodized singular spectrum analysis (APSSA) with Rényi entropy (RE). The adaptive selection of the embedding dimension of the Hankel matrix in APSSA without setting parameters empirically is proposed, and a selection criterion for singular values is established to preprocess the vibration signals of the rolling bearing and enhance the periodic component of the fault. An RE-based threshold value is introduced to further isolate and decouple the impulse segments of the vibration signal in the time domain. By considering the inner raceway fault, outer raceway fault, ball fault, and skidding, a comprehensive simulation model of the compound fault is constructed by the response mechanism of different excited resources. Simulated and experimental data are applied to validate the effectiveness and practicability of the proposed method. The results demonstrate that the RBCF can be identified correctly by the proposed method under strong background noise.

Duality for vector-valued Bergman–Orlicz spaces and little Hankel operators between vector-valued Bergman–Orlicz spaces on the unit ball

Duality for vector-valued Bergman–Orlicz spaces and little Hankel operators between vector-valued Bergman–Orlicz spaces on the unit ball

Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices

Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices

Topology of Hankel matrices and applications

In this paper we first construct a Lie group structure on n×n Hankel matrices over R+ by Hadamard product and then we find its Lie algebra structure and finally calculate dimension of this manifold over R+. Moreover, we discuss topological properties of this manifold using Frobenious norm. We pointed out the relation between Lie group and Lie algebra structures of these matrices by exponential map. It is also shown that the Hadamard product on Hankel matrices over R+ is not bounded by Frobenious norm. Lastly, we provide some applications of these manifolds.

The geometry of the modular bootstrap

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above ∆gap∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\ extrm{gap}}^{\\ast } $$\\end{document}, where c−112<Δgap∗<c12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{c-1}{12}<{\\Delta}_{\ extrm{gap}}^{\\ast }<\\frac{c}{12} $$\\end{document}, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

Schatten class Hankel operators on weighted Bergman spaces induced by regular weights

In this paper, for 1≤p<∞, we provide several descriptions of Schatten p-class Hankel operators Hf and Hf‾ on the weighted Bergman space Aω2, in terms of a certain global and local mean oscillation of the symbol f∈Lω2, provided ω is in a class of regular weights. The approaches applied to rely on several classical methods, and simultaneously rely on a novel but more convenient construction associated with the atomic decomposition of Aω2.

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

The variance and correlations of the divisor function in Fq[T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_q [T]$$\\end{document}, and Hankel matrices

We prove an exact formula for the variance of the divisor function over short intervals in A:=Fq[T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}:= {\\mathbb {F}}_q [T]$$\\end{document}, where q is a prime power; and for correlations of the form d(A)d(A+B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(A) d(A+B)$$\\end{document}, where we average both A and B over certain intervals in A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document}. We also obtain an exact formula for correlations of the form d(KQ+N)d(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(KQ+N) d (N)$$\\end{document}, where Q is prime and K and N are averaged over certain intervals with degN≤degQ-1≤degK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{deg}\\,}}N \\le {{\\,\ extrm{deg}\\,}}Q -1 \\le {{\\,\ extrm{deg}\\,}}K$$\\end{document}; and we demonstrate that d(KQ+N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(KQ+N)$$\\end{document} and d(N) are uncorrelated. We generalize our results to σz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _z$$\\end{document} defined by σz(A):=∑E∣A|A|z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _z (A):= \\sum _{E \\mid A} |A |^z$$\\end{document} for all monics A∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\in {\\mathcal {A}}$$\\end{document}. Our approach is to use the orthogonality relations of additive characters on Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_q$$\\end{document} to translate the problems to ones involving the ranks of Hankel matrices over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_q$$\\end{document}. We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving dk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_k$$\\end{document}.

A multiple waveguide invariant estimation method based on the warping transform and variational mode decomposition

AbstractA multiple waveguide invariant (β) estimation method based on the warping transform and variational mode decomposition (VMD) from one‐dimensional broadband interference sound intensity (BISI) is presented. Firstly, an average β is estimated based on the low‐rank property of the Hankel matrix constructed by BISI. Secondly, the warping principle is used to transform BISI into a warped one. Thirdly, the conventional VMD can be used to effectively decompose the warped BISI. Finally, multiple β can be estimated, respectively. Traditional β estimation methods are based on a two‐dimensional acoustic interference pattern, whose acquisition depends on the horizontal array. This method reduces the data dimensionality of the multivalued β estimation problem without requiring any environmental prior information and is applicable to single hydrophone scenarios. The effectiveness of the presented method is validated by numerical simulation.

Tricomi Continuants

In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some of their basic relations. Then, we obtain a Turan-like inequality, some congruences, some binomial identities (including a Carlitz-like identity), and some relations with the Cayley continuants. Furthermore, we show that the infinite Hankel matrix generated by the Tricomi continuants has an LDU-Sheffer factorization, while the infinite Hankel matrix generated by the shifted Tricomi continuants has an LTU-Sheffer factorization. Finally, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse.

Properties of (C, r)-Hankel Operators and (R, r)-Hankel Operators on Hilbert Spaces

We introduce the operators which are generalizations of Hankel-type operators, called the (C,r)-Hankel operator and (R,r)-Hankel operator on general Hilbert spaces. Our main result is to obtain characterizations for a bounded operator on general Hilbert spaces to be a (C,r)-Hankel operator (or (R,r)-Hankel operator). We also discuss some algebraic properties like boundedness (for |r|≠1) of these operators and the relationship between them. Moreover, some characterizations for the commutativity of these operators are explored.

Identification of Low Order Systems in a Loewner Framework

This paper considers the problem of non-parametric identification of low-order models from time-domain experimental data using a combination of Caratheodory Fejer and Loewner-based interpolation, followed by a Loewner matrix Balanced Reduction (LBR) step. As we show in the paper, the Loewner matrix is an estimator for the trace norm of a system, playing a role similar to the one played by the Hankel matrix. However, utilizing Zolotarev numbers to establish decay rate bounds for singular values reveals that the decay of singular values in the Loewner matrix is considerably faster than that in the Hankel matrix. Thus, Loewner-based methods yield lower order systems, with the same error bound, than comparable ones based on Hankel matrices. The effectiveness of our method is demonstrated through a numerical example.

Deriving Known Particular Solutions of the σ-Commutation Problem (σ ≠ 0, ±1) for a Toeplitz and a Hankel Matrix within a Unified Approach

Deriving Known Particular Solutions of the σ-Commutation Problem (σ ≠ 0, ±1) for a Toeplitz and a Hankel Matrix within a Unified Approach

Exploring Toeplitz, Circulant, and Hankel Matrices: Algebraic Structures and Field Properties with a Focus on Circulant Matrices

This abstract provides an overview of the discussed topics related to Toeplitz matrices, Circulant matrices, Hankel matrices, and their connections to algebraic structures. Additionally, the abstract introduces the exploration of Group, Ring, and Field properties with a focus on the Field Properties of Circulant Matrices. Furthermore, the concept of Eigenvalues in Circulant Matrices is introduced, shedding light on the intrinsic characteristics of these matrices. The abstract concludes by highlighting the visualization aspect through the Graph of column vectors of Circulant Matrices, emphasizing the significance of graphical representations in understanding the matrix properties. Overall, the abstract encapsulates a comprehensive exploration of mathematical concepts and their applications, showcasing the interconnectedness of linear algebra and algebraic structures.

The smallest eigenvalue of large Hankel matrices associated with a semiclassical Laguerre weight

The smallest eigenvalue of large Hankel matrices associated with a semiclassical Laguerre weight

Bayesian High-Rank Hankel Matrix Completion for Nonlinear Synchrophasor Data Recovery

Phasor measurement units (PMUs) provide high temporal-resolution synchrophasor measurements for power system monitoring and control. The frequent data quality issues, such as missing and bad data, prevent the incorporation of synchrophasor data in real-time operations. Most existing data-driven data recovery methods assume the power system dynamics can be approximated by a linear dynamical system, and the recovery performance degrades significantly when the power system is experiencing nonlinear dynamics during significant events. This paper proposes a data-driven Bayesian nonlinear synchrophasor data recovery method (Ba-NSDR) that can recover a consecutive time period of simultaneous data losses or errors across all channels, even when the underlying system is highly nonlinear. The idea is to lift the Hankel matrix of the spatial-temporal synchrophasor data to a higher dimension such that the lifted Hankel matrix is low-rank in that space and can be processed with the kernel trick. Our proposed Bayesian method then infers the probabilistic distributions of synchrophasor from the partial observations. Some distinctive features of Ba-NSDR include an uncertainty index to measure the accuracy of the recovery result and the robustness to parameter selections. Our method is verified on both synthetic and recorded event datasets.

Data Driven Positive Subspace System Identification

This paper considers the data-driven subspace system Identification for positive systems. The application of subspace system Identification (SSID) method to collected data of positive systems does not guarantee the positivity of the identified system. Thus, two approaches are proposed to obtain the state space parameters of the positive system using the data collected from its input and output. The first approach offers a procedure based on nonnegative least square algorithm applied to data-driven state space representation. The second approach applies a nonnegative matrix factorization to the data collected Hankel matrix followed by generalized realization algorithm. The effectiveness of two approaches in positive SSID are demonstrated by numerical examples.