Hankel Matrix Research Articles

The Nehari Problem for the Paley–Wiener Space of a Disc

There is a bounded Hankel operator on the Paley–Wiener space of a disc in {mathbb {R}}^2 which does not arise from a bounded symbol.

On the spectral properties of real antitridiagonal Hankel matrices

Abstract In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.

Construction of a Random Matrix Model of Curriculum System Based on Digital Technology in the Context of Smart Education

This paper adopts the method of digital technology fused with a random matrix to conduct in-depth research and analysis on the curriculum system under the context of smart education and designs a corresponding model for practical use. Firstly, we analyze the status of digital learning research, take the digital teaching model based on independent inquiry as to the basis of teaching design, and carry out digital-based teaching experiments under the premise of fully examining the actual teaching conditions. The digital learning platform is designed to meet the teaching model, platform design, platform application, and implementation of the school, and the digital learning platform is designed and developed through PHP technology. Then, the matrix filling algorithm is used to fill in some of the missing data in the observation matrix to obtain the complete data matrix, and the Hankel matrix inverse transformation is used to finally obtain the reconstructed original signal, thereby forming the ability of innovative works. It is an important aspect of cultivating students’ innovative spirit and practical ability in the development of educational informatization. Ultimately, students use the platform to try out and be able to learn the knowledge. The experimental class and the control class experimented with a quasi-experimental research method, the experimental class used a digital teaching mode, and the control class used a traditional IT teaching mode. The digital learning ability of students was measured by a digital learning ability measurement scale before and after the experiment, and then the data before and after the experiment were analyzed by horizontal and vertical comparison to verify whether the digital teaching model can effectively cultivate junior high school students compared with the traditional teaching model. The digital learning ability of junior high school students was verified.

Accelerated MRI Reconstruction With Separable and Enhanced Low-Rank Hankel Regularization.

Magnetic resonance imaging serves as an essential tool for clinical diagnosis, however, suffers from a long acquisition time. Sparse sampling effectively saves this time but images need to be faithfully reconstructed from undersampled data. Among the existing reconstruction methods, the structured low-rank methods have advantages in robustness to the sampling patterns and lower error. However, the structured low-rank methods use the 2D or higher dimension k-space data to build a huge block Hankel matrix, leading to considerable time and memory consumption. To reduce the size of the Hankel matrix, we proposed to separably construct multiple small Hankel matrices from rows and columns of the k-space and then constrain the low-rankness on these small matrices. This separable model can significantly reduce the computational time but ignores the correlation existed in inter- and intra-row or column, resulting in increased reconstruction error. To improve the reconstructed image without obviously increasing the computation, we further introduced the self-consistency of k-space and virtual coil prior. Besides, the proposed separable model can be extended into other imaging scenarios which hold exponential characteristics in the parameter dimension. The in vivo experimental results demonstrated that the proposed method permits the lowest reconstruction error with a fast reconstruction. The proposed approach requires only 4% of the state-of-the-art STDLR-SPIRiT runtime for parallel imaging reconstruction, and achieves the fastest computational speed in parameter imaging reconstruction.

Fault Diagnosis of Rotating Equipment Bearing Based on EEMD and Improved Sparse Representation Algorithm

Aiming at the problem that the vibration signals of rolling bearings working in a harsh environment are mixed with many harmonic components and noise signals, while the traditional sparse representation algorithm takes a long time to calculate and has a limited accuracy, a bearing fault feature extraction method based on the ensemble empirical mode decomposition (EEMD) algorithm and improved sparse representation is proposed. Firstly, an improved orthogonal matching pursuit (adapOMP) algorithm is used to separate the harmonic components in the signal to obtain the filtered signal. The processed signal is decomposed by EEMD, and the signal with a kurtosis greater than three is reconstructed. Then, Hankel matrix transformation is carried out to construct the learning dictionary. The K-singular value decomposition (K-SVD) algorithm using the improved termination criterion makes the algorithm have a certain adaptability, and the reconstructed signal is constructed by processing the EEMD results. Through the comparative analysis of the three methods under strong noise, although the K-SVD algorithm can produce good results after being processed by the adapOMP algorithm, the effect of the algorithm is not obvious in the low-frequency range. The method proposed in this paper can effectively extract the impact component from the signal. This will have a positive effect on the extraction of rotating machinery impact features in complex noise environments.

On *-Convergence of Schur–Hadamard Products of Independent Nonsymmetric Random Matrices

Abstract Let $\{x_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$ and $\{y_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix $X_n = ((x_{i - j}))_{1 \le i, j \le n}$ and a Hankel matrix $Y_n = ((y_{i + j}))_{1 \le i, j \le n}$, and let $M_n = X_n \odot Y_n$ be their elementwise/Schur–Hadamard product. In this article, we show that almost surely, $n^{-1/2}M_n$, as an element of the *-probability space $(\mathcal {M}_n({\mathbb {C}}), \frac {1}{n}\text {tr})$, converges in *-distribution to a circular variable. With i.i.d. Rademacher entries, this construction gives a matrix model for circular variables with only $O(n)$ bits of randomness. We also consider a dependent setup where $\{x_{\alpha }\}$ and $\{y_{\beta }\}$ are independent strongly multiplicative systems (à la Gaposhkin [7]) satisfying an additional admissibility condition, and have uniformly bounded moments of all orders—a nontrivial example of such a system being $\{\sqrt {2}\sin (2^n \pi U)\}_{n \in {\mathbb {Z}}_+}$, where $U \sim \textrm {Uniform}(0, 1)$. In this case, we show in-expectation and in-probability convergence of the *-moments of $n^{-1/2}M_n$ to those of a circular variable. Finally, we generalise our results to Schur–Hadamard products of structured random matrices of the form $X_n = ((x_{L_X(i, j)}))_{1 \le i, j \le n}$ and $Y_n = ((y_{L_Y(i, j)}))_{1 \le i, j \le n}$, under certain assumptions on the link-functions$L_X$ and $L_Y$, most notably the injectivity of the map $(i, j) \mapsto (L_X(i, j), L_Y(i, j))$. Based on numerical evidence, we conjecture that the circular law $\mu _{\textrm {circ}}$, that is, the uniform measure on the unit disk of ${\mathbb {C}}$, which is also the Brown measure of a circular variable, is in fact the limiting spectral measure (LSM)of $n^{-1/2}M_n$. If true, this would furnish an interesting example where a random matrix with only $O(n)$ bits of randomness has the circular law as its LSM.

Estimation of the Complexity of a Finite Mixture Distribution: From Well- to Less Known Methods

Mixture models occur in numerous settings including random and fixed effects models, clustering, deconvolution, empirical Bayes problems and many others. They are often used to model data originating from a heterogeneous population, consisting of several homogeneous subpopulations, and the problem of finding a good estimator for the number of components in the mixture arises naturally. Estimation of the order of a finite mixture model is a hard statistical task, and multiple techniques have been suggested for solving it. We will concentrate on several methods that have not gained much popularity yet deserve the attention of practitioners. These can be categorized into three groups: tools built upon the determinant of the Hankel matrix of moments of the mixing distribution, minimum distance estimators, likelihood ratio tests. We will address theoretical pillars underlying each of the methods, provide some useful modifications for enhancing their performance and present the results of the comparative numerical study that has been conducted under various scenarios. According to the results, none of the methods proves to be a “magic pill”. The results uncover limitations of the techniques and provide practical hints for choosing the best-suited tool under specific conditions.

Coordinate sections of generic Hankel matrices

The core of this work is to understand the structural properties of certain coordinate sections of the generic square Hankel matrix over a field k of characteristic zero. In particular, one looks closely at the Jacobian ideal of the determinant, its Hessian and the ideal of submaximal minors of the matrix. In contrast to coordinate sections of the fully generic square matrix and the symmetric generic matrix, here the Hessian is nonzero, hence the related polar map is dominant. In the case of the fully generic Hankel matrix one establishes some of the conjectures stated in a previous paper by the second and fourth authors. A major question in this connection is whether the Jacobian ideal is of linear type, a result that would trigger similar results for the coordinate sections considered here. At the moment this is only known for the so-named sub-Hankel coordinate section.

Fault diagnosis of rotating machinery in a noisy environment based on modified general normalized sparse filtering

Intelligent fault diagnosis for mechanical condition monitoring has achieved a great deal of success in recent years, but most of the research is carried out in experimental environments. Vibration signals collected in real scenarios usually have strong noise interference, which significantly reduces fault classification capability and seriously affects the accuracy and robustness of classification. This paper proposes modified general normalized sparse filtering (MGNSF) with strong noise adaptability for rotating machinery fault diagnosis without any time-consuming denoising preprocessing, in which generalized normalization, weight and feature normalization, and the Hankel matrix. Diagnostic performance is studied with the change of normalization parameters and signal noise ratio. Weight and feature normalization can improve the distribution between features. The proposed algorithm is validated using two rolling bearing datasets. The experimental results show that MGNSF can extract the features of a faulty bearing under stronger noise interference and has strong noise adaptability.

Fast NMR spectroscopy reconstruction with a sliding window based Hankel matrix

Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most promising analytical chemistry techniques, although it takes a long time to acquire data. Non-uniform sampling (NUS) is an effective way to reduce the sampling time, but faithful reconstruction methods are needed. The low rank Hankel matrix (LRHM) approach uses the low rank constraint to obtain high-quality spectra from NUS signals, but the reconstruction has a considerable time overhead. In this work, we propose a sliding window based low rank Hankel matrix approach to speed up the spectra reconstruction from NUS signals. Using the sliding window to construct a matrix can effectively reduce the size of the Hankel matrix for faster reconstructions. To further decrease the reconstruction time, parallel computation is applied in the proposed approach. The experiments on both synthetic data and realistic data demonstrate that the reconstruction speed of the proposed method is the fastest among compared methods without sacrificing the quality of spectra.

Self-adjoint operators associated with Hankel moment matrices

In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments (mn) of a positive measure with infinite support on the real line. If mn=o(1) Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations.The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences.We do this not only in the case mn=o(1) studied by Yafaev but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy.

Schur–Nevanlinna parameters, Riesz bases, and compact Hankel operators on the model space

We study Riesz bases/Riesz sequences of reproducing kernels in the model space Kθ in connection with the corresponding Schur–Nevanlinna parameters and functions. In particular, we construct inner functions with given Schur–Nevanlinna parameters at a given sequence Λ such that the corresponding systems of projections of reproducing kernels in the model space are complete/non-complete. Furthermore, we give a compactness criterion for Hankel operators with symbol θB‾, where θ is an inner function and B is an interpolating Blaschke product, and use this criterion to describe the Riesz bases KΛ,θ with limλ∈Λ,|λ|→1⁡θ(λ)=0.

Singular Value Decomposition Based Pilot Protection for Transmission Lines With Converters on Both Ends

The modular multilevel converter-based high voltage direct current transmission (MMC-HVDC) technology has become an effective integrating method for large-capacity offshore wind farms. However, if AC side faults occur between the MMC and the wind farm, conventional current differential protection methods based on phasor measurements might not detect the large fault current at the wind farm and the MMC, resulting in poor protection performance. In the worst case, MMC blocking may occur during faults near the MMC station, causing relay refusal. The fault currents provided by both converters are analyzed in detail. Due to the fault-ride-through control objectives, it is proved that the singularity of fault current between wind farms and MMC is different. Therefore, a novel singular value decomposition (SVD) based pilot protection is proposed to solve this problem. A Hankel matrix is established for the short-circuit current at each end of the AC line. The Hankel matrix rapidly detects the singularity of the signal and its derivative, and the SVD method uses a short time window to extract the waveform features. This approach enables the detection of the fault features before MMC blocking in the worst-case fault scenario and prevents the incorrect fault identification caused by the minor amplitude of fault current provided by the converters. A detailed model of an offshore wind farm connected to the MMC-HVDC system is established in the real-time digital simulator (RTDS). A protection device compliant with industry standards is used to implement the SVD-based protection algorithm. Hardware-in-loop dynamic experimental tests are conducted to verify the performance of the proposed protection method. The experimental results demonstrate that the SVD-based protection method has excellent performance during various fault scenarios.

A Novel Reinforced Deep RNN–LSTM Algorithm: Energy Management Forecasting Case Study

In this article, a new hybrid deep learning (DL) algorithm is developed to make a computer-assisted forecasting energy management (EM) system. Applying the Copula function, the Hankel matrix is created for processing gathered automatic metering infrastructure (AMI) load information in the smart network. This processing of the data results in model optimization through the suggested new pooling-based deep neural network (PDNN). Through increased size and variation of AMI data, the suggested PDNN reduces overfitting issues during testing and training. The real-time AMI southern grid data of Tamil Nadu electricity is used as the benchmark. The suggested DL model performs better than the traditional EM forecasting techniques in both mean absolute error and accuracy by 12.7% and 9.5%, respectively.

Internally Hankel $k$-Positive Systems

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of $k$-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most $k-1$. For $k=1$, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of $k$-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for $k=1$, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of $k$-positivity by analyzing internally Hankel $k$-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are $k$-positive. We show that the existing notions of external Hankel and internal $k$-positivity are subsumed under internal Hankel $k$-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first $k$ compound systems. As such, this class provides new means to verify external Hankel $k$-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel $k$-positive realizations.

Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems.

The strong truncated Hamburger moment problem with and without gaps

The strong truncated Hamburger moment problem (STHMP) of degree (−2k1,2k2) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R, such that βi=∫xidμ(−2k1≤i≤2k2). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. Then, using the equivalence with the STHMP of degree (−2k,2k), we obtain the solution of the 2–dimensional truncated moment problem (TMP) of degree 2k with variety xy=1, first solved by Curto and Fialkow [22]. Our addition to their result is the fact previously known only for k=2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k1,2k2) with one missing moment in the sequence, i.e., β−2k1+1 or β2k2−1, which also gives the solution of the TMP with variety x2y=1 as a special case, first studied by Fialkow in [33].

DOA estimation algorithm based on spread spectrum sequence in low signal-to-noise ratio

Spread spectrum communication is a common communication method in underwater communication. Based on the space-time processor received by the array, it can filter the signals arriving along each path separately. Combined with the diversity of space-time clusters, it can effectively improve the communication system’s reliability. The core problem of the space-time processor is the direction of arrival (DOA) and signal source number estimation. Based on the good self-coherence of the spread spectrum sequence, this paper proposes a multiple signal classification algorithm (MUSIC) for accurate DOA estimation. However, since the MUSIC algorithm uses the received signal’s covariance matrix for DOA estimation, the number of sources needs to be predicted in advance. Under a low signal-to-noise ratio (SNR), the signal eigenvalues and the noise eigenvalues of the covariance matrix differ slightly, which makes signal source number estimation difficult. To address this issue, a singular value decomposition method using the delay structure information of the array element is proposed to estimate the number of sources of the spreading sequence under a low SNR. The method proposed in this paper can well estimate the DOA of the signal under a low SNR. Meanwhile, there is no need to convert the signal to the individual sub-bands, which effectively reduces the calculation overhead. At the same time, the Hankel matrix is used to solve the problem that the MUSIC algorithm cannot accurately estimate the number of signal sources under the condition of low SNR. Compared with the conventional algorithm, the Hankel matrix can more accurately estimate the number of signal sources in the case of low SNR. Through simulation experiments, the effectiveness of our DOA estimation algorithm is validated under a low SNR.

Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators

Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)>ess inf|ψ|, which improves a result from [15].

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.