Hankel Matrix Research Articles

Commutativity of Hankel and Toeplitz operators on the Hardy space of the n-torus

We consider Hankel and Toeplitz operators on H2(Tn), the Hardy space of the n-torus Tn. Given symbols φ and ψ in L∞(Tn) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator Hψ,n and the Toeplitz operator Tφ,n to commute. We then extend the study to the more general situation where no assumptions are imposed on φ, and provide new, non-trivial necessary conditions for the commutativity of Hψ,n and Tφ,n. We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting.

System identification based on the cross Gramian

In order to make mechanical structures more reliable, research is being carried out into methods of detecting and localizing damage to structures, known as Structural Health Monitoring (SHM). One method for continuously monitoring a structure is called State Projection Estimation Error (SP2E) and was developed at the University of Applied Sciences Leipzig. This method is based on state space systems, which are generated by output-only system identification. It is state-of-the-art to create a stabilization diagram to choose the best model order. Since the monitoring process is continuous, a state space model is created every 10 minutes, and therefore it is not possible to look at a stabilization diagram. Thus, the system identification process has to be automated. A novel approach developed in the SPP 100+ research program takes place via the theory of the cross Gramian. The cross Gramian is defined by the Hankel operator, which maps the past input to the future output. This Gramian indicates the interaction between the states and the input/output of a system and can be seen as the energy capacity of a system. By comparing the power of the measurement with the identified energy capacity, the model order selection can be automated. There are limits to the existence of the cross Gramian for multiple-input, multiple-output (MIMO) systems. The system has to be square, stable and symmetric. The covariance driven system identification never provides a symmetric system. A way to create a symmetric system out of the asymmetric measurement data is presented by the use of a symmetrizer, such that the cross Gramian can always be computed. With this symmetrizer a method to estimate the system matrix A by a stable Lyapunov equation, which is classically computed by a numerically unstable pseudo inverse, is presented.

Joint moments of higher order derivatives of CUE characteristic polynomials II: structures, recursive relations, and applications

In a companion paper (Keating and Wei 2023 Int. Math. Res. Not. 2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III ′ equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.

Schatten class Hankel operators on doubling Fock spaces and the Berger-Coburn phenomenon

Using the notion of integral distance to analytic functions, we give a characterization of Schatten class Hankel operators acting on doubling Fock spaces on the complex plane and use it to show that for f∈L∞, if Hf is Hilbert-Schmidt, then so is Hf¯. This property is known as the Berger-Coburn phenomenon. When 0<p≤1, we show that the Berger-Coburn phenomenon fails for a large class of doubling Fock spaces. Along the way, we illustrate our results for the canonical weights |z|m when m>0.

Truncated Hankel operators on model spaces

The paper by Sarason in 2007 initiated and inspired the study of truncated Toeplitz operators which are the compression of Toeplitz operators to coinvariant subspace of H2. We study the truncated Hankel operators which are the compression of Hankel operators to coinvariant subspace of H2. Several parallel characterizations and properties of such operators are obtained. In addition, boundedness, finite rank and compactness of truncated Hankel operators were characterized in terms of their symbols.

Factoring determinants and applications to number theory

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of [Formula: see text]-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove that, in certain cases, these unitary averages factor as polynomials into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the “swap” terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from random matrix theory.

Behaviour Analysis of Modeling and Model Evaluating Methods in System Identification for a Multiprocess Station

Systems are designed to perform specific task by giving certain input which produces the required output in an orderly manner known as process. The input, output, and the state variables should be known that will help in interacting with the system. The relation between these variables can be brought out by building a model that resembles or expresses the original performance of the system. The parameters of the model are estimated using the least squares approximation, maximum likelihood, maximum log-likelihood, and Bayesian parameter estimation methods by utilizing the experimental data from the multiprocess station. The selected parameters are converted to nine different transfer function models that represent the given dynamic system. The models framed are analyzed by the criterion curve technique using seven criterion functions evaluating the fitness of the model. Order of the model is found from Hankel matrix representation methods such as singular value decomposition and determinant method. Response of the models is compared with the original response to choose the best fit model by calculating ISE standard. All the above methods are used to model the system without physical and theoretical laws which is known as system identification.

Data-driven economic MPC with asymptotic stability and strong duality verification using Hankel matrix

We consider the problem of dynamic regulation with an economic cost function to control unknown linear systems, in which improving the economic performance and guaranteeing the stability of economical optimal equilibrium point are control objectives. A data-driven economic MPC scheme is presented using measured input-output trajectories without a prior system identification step. Our method uses Hankel matrices which include one input-output data trajectory for prediction in economic MPC, while persistently exciting of the input generating the data is needed. One of the novelties of the presented framework is directly verifying the strong duality property from input-output trajectory with the general cost function, considered as the supply rate. This is used to find a Lyapunov function for data-driven economic MPC. Under the strong duality assumption, asymptotic stability of the economical optimal equilibrium point for the closed-loop system with terminal equality constraint is guaranteed. The proposed data-driven economic MPC approach needs only persistently exciting data trajectory along with an upper bound on the system order and need no model description and no online parameter estimation. The proposed scheme applicability compared to the existing model-based economic MPC and data-driven MPC is illustrated for continuous stirred tank reactor (CSTR) and a numerical example and the robustness of the proposed scheme is evaluated in the case of measurement noise, as well as nonlinear model for CSTR system.

Adaptive fast nonlinear blind deconvolution based on nonuniform particle swarm optimization for the rolling bearing fault diagnosis

Although superior fault extraction performance is possessed by fast nonlinear blind deconvolution (FNBD) through the introduction of penalty terms and nonlinear features, its wide application is limited by the complexity and diversity of parameter selection. The fault extraction performance of FNBD is impacted by the contingency of parameter selection. To reduce the prior experience dependence of FNBD and improve the performance of fault extraction, an end-to-end adaptive fast nonlinear blind deconvolution (AFNBD) is proposed. First, the proposed nonuniform particle swarm optimization (NPSO) is employed to assign parameters to be optimized for each particle to construct its Hankel matrix, nonlinear features and objective functions. Secondly, Gaussian fitting is applied to modify the filters. Then, NPSO is utilized to update relevant parameters and the corresponding filter of each particle to explore the optimum filter. Finally, simulations and experiments verify the effectiveness of AFNBD and the results indicate that AFNBD is a powerful tool for incipient fault feature extraction.

Interference mitigation for FMCW radar via chirp rate estimation and signal separation

As a prevalent system in radar technology, the Frequency modulated continuous wave (FMCW) radar faces significant challenges in interference. This paper proposes a three-step interference mitigation strategy considering the time–frequency (TF) features of interference and time-domain properties of beat signals. Initially, the process involves statistical analysis of chirp rates from high-energy TF points, estimating the matched chirp rates of interference signals. Subsequently, the improved multisynchrosqueezing chirplet transform (IMSSCPT) is suggested for high-resolution TF characterization of interference signals. Utilizing the ridge extraction method, we accurately detect the TF region of interference signals. This precise localization enables the gradual separation of the reconstructed interference from the received signals. Finally, we perform parallel singular value decomposition on Hankel matrices, which are stacked from the mitigated subsequences to obtain the beat subspace and reconstruct the beat signal to reduce the negative effect of noise. A series of experimental comparisons of the proposed algorithm with existing methods have validated its exceptional performance, particularly under conditions of high-level noise.

Multipliers and equivalence of functions, spaces, and operators

By interpreting the well-known Brown–Halmos theorem for Toeplitz operators in terms of multipliers, we formulate a Brown–Halmos analogue for the product of generalized Toeplitz operators, defined as compressions of multiplication operators to closed subspaces of L2(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2({\\mathbb {T}})$$\\end{document}. We use this to define equivalences between two operators in that class by means of multipliers between the spaces where they act. Necessary and sufficient conditions for such an equivalence to be unitary or a similarity are established. The results are applied to Toeplitz and Hankel operators, truncated Toeplitz operators, and dual truncated Toeplitz operators.

An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line

We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory, we prove the genericity of turbulent solutions of the cubic Szegő equation on the real line.

Hankel operators on Lp(ℝ+) and their p-completely bounded multipliers

Hankel operators on Lp(ℝ+) and their p-completely bounded multipliers

Mixed product of Hankel and Toeplitz operators on Fock-Sobolev spaces of negative orders

Mixed product of Hankel and Toeplitz operators on Fock-Sobolev spaces of negative orders

Eigenvalue estimates for Fourier concentration operators on two domains

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers, for the first time, non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel’s work on one dimensional intervals arXiv:1502.04404v1. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 &gt; s &gt; 1 0&gt;s&gt;1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 &gt; s &gt; 1 0&gt;s&gt;1 .

On the Hankel matrices of finite rank and generalized Fibonacci sequences

On the Hankel matrices of finite rank and generalized Fibonacci sequences

Determination of Dynamic Characteristics of Lattice Structure Using Dynamic Mode Decomposition

Abstract In this work, dynamic mode decomposition (DMD) was applied as an algorithm for determining the natural frequency and damping ratio of viscoelastic lattice structures. The algorithm has been developed based on the Hankel alternative view of Koopman (HAVOK) and DMD . In general, the Hankel matrix is based on time-delay embedding, which is meant for the hidden variable in a time-series data. Vibration properties of a system could be then estimated from the eigenvalues of the approximated Koopman operator. Results of the proposed algorithm were firstly validated with those of the traditional discrete Fourier transform (DFT) approach and half-power bandwidth (HPBW) by using an analytical dataset of multi-modal spring-mass-damper system. Afterward, the algorithm was further used to analyze dynamic responses of viscoelastic lattice structures, in which data from both experimental and numerical finite element (FE) model were considered. It was found that the DMD-based algorithm could accurately estimate the natural frequencies and damping ratios of the examined structures. In particular, it is beneficial to any dataset with limited amounts of data, whereby experiments or data gathering processes are expensive.

The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

In this paper, we study the large N behavior of the smallest eigenvalue λN of the (N+1)×(N+1) Hankel matrix, HN=(μj+k)0≤j,k≤N, generated by the γ dependent Jacobi weight w(z,γ)=e−γzzα(1−z)β,z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials PN(z), z∈C﹨[0,1], with the weight w(z,γ)=e−γzzα(1−z)β. Using the polynomials PN(z), we obtain the theoretical expression of λN, for large N. We also display the smallest eigenvalue λN for sufficiently large N, computed numerically.

Seismic noise attenuation method based on low‐rank adaptive symplectic geometry decomposition

AbstractThe basic assumption of low‐rank methods is that noise‐free seismic data can be represented as a low‐rank matrix. Effective noise reduction can be achieved through the low‐rank approximation of Hankel matrices composed of the data. However, selecting the appropriate rank parameter and avoiding expensive singular value decomposition are two challenges that have limited the practical application of this method. In this paper, we first propose symplectic geometric decomposition that avoids singular value decomposition. The symplectic similarity transformation preserves the essence of the original time sequence as well as the signal's basic characteristics and maintains the approximation of the Hankel matrix. To select an appropriate rank, we construct the symplectic geometric entropy according to the distribution of eigenvalues and search for high‐contributing eigenvalues to determine the needed rank parameter. Therefore, we provide an adaptive approach to selecting the rank parameter by the symplectic geometric entropy method. The synthetic examples and field data results show that our method significantly improves the computational efficiency while adaptively retaining more effective signals in complex structures. Therefore, this method has practical application value.