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In this paper, we evaluate the following three symmetric Toeplitz determinants: $$\begin{align*} &{\rm det}{[|j-k|+\delta_{jk}]_{1\leq j,k\leq n}},\ {\rm det}[F_{|j-k|}+\delta_{jk}]_{1\le j,k\le n} \quad {\rm and}\\ &{\rm det}\left[\genfrac{(}{)}{}{}{|j-k|}{3}-\delta_{jk}\right]_{1\le j,k\le n}, \end{align*} $$ det [ | j − k | + δ jk ] 1 ≤ j , k ≤ n , det [ F | j − k | + δ jk ] 1 ≤ j , k ≤ n and det [ ( | j − k | 3 ) − δ jk ] 1 ≤ j , k ≤ n , where $ \delta _{jk} $ δ jk is the Kronecker delta, $ (F_i)_{i\ge 0} $ (F i) i ≥ 0 is the Fibonacci sequence, and $ \left (\frac {\cdot }3\right) $ (⋅ 3) is the Legendre symbol. [ABSTRACT FROM AUTHOR]

By considering the tiling of an N-board (a linear array of N square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers $ s_n $ s n (where $ s_{n}=\sum _{i=1}^q v_i s_{n-m_i} $ s n = ∑ i = 1 q v i s n − m i , $ s_0=1 $ s 0 = 1 , $ s_{n \lt 0}=0 $ s n < 0 = 0 , where $ v_i $ v i and $ m_i $ m i are positive integers and $ m_1 \lt \cdots \lt m_q $ m 1 < ⋯ < m q ) each raised to an arbitrary non-negative integer power. A $ (w,g;m) $ (w , g ; m) -comb is a tile composed of m rectangular sub-tiles of dimensions $ w\times 1 $ w × 1 separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating $ s_n^2 $ s n 2 for the cases q = 2, $ v_i=1 $ v i = 1 , $ m_1=M $ m 1 = M , $ m_2=m+1 $ m 2 = m + 1 for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are $ -2 $ − 2 , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers. [ABSTRACT FROM AUTHOR]

We introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems. [ABSTRACT FROM AUTHOR]

We consider a symmetric Toeplitz matrix completion problem, of which the matrix possesses special row and column structures. It has wide applications in diverse areas and is well-known to be computationally NP-hard. This note presents an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on conclusions from the trigonometric moment problem and the semi-infinite problem. We prove that the upper bound is less than twice the number of the active constraints of the associated semi-infinite problem. Moreover, it is less than twice the number of linear constraints of the problem. [ABSTRACT FROM AUTHOR]

We find an algorithmic procedure that enables the computation and description of the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper triangular Toeplitz matrix equation. [ABSTRACT FROM AUTHOR]

We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study the structure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021]. [ABSTRACT FROM AUTHOR]

Using some concepts and tools from Functional Analysis, we express explicit forms of 'eigenvectors' of two Toeplitz operators: An operator of multiplicity one, which appears e.g. in Autoregressive Process, and a banded operator of high multiplicity. Then we investigate the approximate eigenvectors of their associated Toeplitz matrices of large sizes, and after some numerical experiments, the paper concludes with an application in the principal component analysis method for dimension reduction. [ABSTRACT FROM AUTHOR]

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261–278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutiérrez–Gutiérrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885–891], Öteleş and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448–10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000–1003]. [ABSTRACT FROM AUTHOR]

Tridiagonal Toeplitz P-matrices, M-matrices and totally positive matrices are characterized. For some classes of tridiagonal matrices and tridiagonal Toeplitz matrices, it is shown that many algebraic computations can be performed with high relative accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix is proved to be clustered around 1, which can guarantee the superlinear convergence rate of the proposed method. Numerical experiments are carried out to demonstrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we determine the approximate eigenvalue bound state solution of the radial Schrodinger equation in three-dimension in the presence of the Aharonov-Bohm flux field with two-term diatomic molecular potential in point-like global monopole (PGM) defect. We analyse the effects of factors, such as topological defects, background curvature associated with topological defects as well as the potential. We also show that the energy levels shift due to the presence of the magnetic flux which gives us an analogue of the Aharonov-Bohm effect for the bound state. Finally, we apply the quantum system for some known potential models, such as q-deformed Hulthen-type potential and Manning-Rosen potential, and discussed the effects of various factors on the eigenvalue solutions. [ABSTRACT FROM AUTHOR]

A Pascal-like matrix is constructed whose row entries are the terms of the modified Hermite polynomials of two variables. The multiplication of the two Pascal-like matrices and the power and inverse of the Pascal-like matrix have very similar results to the well-known Binomial Matrix, and the factorization of the Pascal-like matrix has also been given, which is completely similar to the factorization of Binomial Matrix. Furthermore, two simple examples are given to show the application of the power and factorization properties of the Pascal-like matrix. Finally, a generalization of the Pascal-like matrix is given and some combinatorial identities are obtained such as Tepper-like identity. [ABSTRACT FROM AUTHOR]

Given disjoint non-empty subsets S and T of { 1 , ... , n − 1 } , a digraph D with the vertex set { 1 , 2 , ... , n } is called a directed Toeplitz graph provided the arc i → j occurs if and only if i − j ∈ S or j − i ∈ T. We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with n ≥ 6 vertices has exponent at least 3 and for each n ≥ 6 , there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that exp ⁡ (D) ≤ (n − 1) 2 + 1 for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained. [ABSTRACT FROM AUTHOR]

In this work, we present and prove the explicit formula for the determinant of a class of n × n nonsymmetric Toeplitz matrices. Setting up one of the nonzero subdiagonals to zero results in special pentadiagonal Toeplitz matrices, whose determinant formulas are conjectured by Anđelić and Fonseca in Anđelić [Some determinantal considerations for pentadiagonal matrices. Linear Multilinear Algebra. 2020. DOI:]. By using the explicit formula with this setting, the conjectures are therefore proved. [ABSTRACT FROM AUTHOR]

This paper contains interesting facts regarding the powers of odd ordered special circulant magic squares along with their magic constants. It is shown that we always obtain circulant semi-magic square and special circulant magic square in the case of even and odd positive integer powers of these magic squares respectively. These magic squares will also take a particular form in the case of both even and odd positive integer powers. Moreover, they are singular under some conditions. [ABSTRACT FROM AUTHOR]

In Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. we have considered the problem about asymptotic formulas for all eigenvalues of T n (a) , as n goes to infinity, assuming that a is a specific model symbol with a unique zero of order 4. In this paper, we continue our investigation and we explore the case where a is a more general real-valued rational symbol with a unique zero of order 4. It should be noted that we apply a different method than the one used in Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. This method coming from works Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J Math Anal Appl. 2015;422(2):1308–1334 and Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017. p. 179–212, where it is considered the class of all symbols having zeros of second order and one can reduce the problem to asymptotic analysis of a nonlinear equation. As well, we construct uniform asymptotic expansions for all eigenvalues, which allow us to precise the classical results of Widom and Parter for first and very last eigenvalues. [ABSTRACT FROM AUTHOR]

The matrix Gaussian series refers to a sum of fixed matrices weighted by independent standard normal variables and plays an important in various fields related to probability theory. In this paper, we present the dimension-free tail bounds and expectation bounds for the largest singular value (LSV) of matrix Gaussian series, respectively. By using the resulting bounds, we compute the expectation bounds for LSVs of Gaussian Wigner matrix and Gaussian Toeplitz matrix, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we will consider matrices with entries in the space of operators (H), where H is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in (H). We shall also introduce matricial versions with operator entries of classical spaces of holomorphic functions such as H∞() and A() when dealing with upper triangular matrices. [ABSTRACT FROM AUTHOR]

In this paper the model of experiment in which several characteristics are measured or the measurements are repeated in time, is considered. It is assumed that the covariance matrix of the characteristics/time points is unknown, but has the banded Toeplitz structure. The problem of estimating such a covariance structure was considered by e.g. Cui et al. (2016), Filipiak et al. (2018). We discuss two methods that aim to deal with singularity and numerical ill-conditioning of the estimators of covariance matrix under high-dimensional regime. The method of estimation of banded Toeplitz covariance matrix based on a shrinking, described by Ledoit and Wolf (2004) for unstructured covariance matrices is proposed. The estimators obtained with the use of the proposed method are then compared with the estimators obtained by the projection of sample covariance matrix onto the asymptotic cone of the nonnegative definite Toeplitz matrices (Filipiak et al. 2018). For this purpose simulation studies concerning bias and risk are conducted for several sets of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we show that the lower companion matrix associated with the monic polynomial p (λ) = p 0 + p 1 λ + ⋯ + p n − 1 λ n − 1 + λ n over a ring with unity is core invertible if and only if there exist solutions x and y to (i) (p 0 x) ∗ = p 0 x , (ii) y p 0 = 0 = p 0 y , (iii) (x , y) ( p 0 p 1 ) = − 1 , (iv) (1 + p 0 x) (p 1 x , 1 + p 1 y) = (0 , 0). In addition, some methods of characterizing and computing the core inverse of a Hankel matrix and a Toeplitz matrix over a field are presented. Moreover, the expressions of core inverses of these matrices are presented. [ABSTRACT FROM AUTHOR]

In this article, we introduced the concept of a ring of stable range 2 which has square stable range 1. We proved that a Hermitian ring R which has (right) square stable range 1 is an elementary divisor ring if and only if R is a duo ring of neat range 1. And we found that a commutative Hermitian ring R is a Toeplitz ring if and only if R is a ring of (right) square range 1. We proved that if R be a commutative elementary divisor ring of (right) square stable range 1, then for any matrix A ∈ M 2 (R) one can find invertible Toeplitz matrices P and Q such that PAQ = ( e 1 0 0 e 2 ) , where e1 is a divisor of e2. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit formula for the element of the inverse where is a band Toeplitz matrix with left bandwidth s and right bandwidth r. The formula involves determinants, , whose elements are the discrete Fourier transform of where f is the symbol of . [ABSTRACT FROM AUTHOR]

In this paper we study the Fock-type space of analytic functions on the plane, invariant with respect to rotations of the plane by , with weight generated in a special way from the solutions of the higher-order Bessel equation. We give an explicit construction of the Bergman kernel for this space; and we prove that the operator of multiplication by is adjoint in this space to the higher-order Bessel operator. These instruments are used to investigate, in this new setting, some standard properties of operators in this space, including the basic properties of Toeplitz and Hankel operators, the uncertainty property and extremal functions for some operators. [ABSTRACT FROM AUTHOR]

We give a characterization of binormal operator matrices, and in particular, a description of binormal Toeplitz matrices for . Non-trivial examples of binormal matrices that are not normal are also provided. [ABSTRACT FROM AUTHOR]

An expression including symmetric polynomials for the determinant of one type of generalized Vandermonde matrix is presented leading to a handy formula to compute determinants of general Toeplitz band matrices. [ABSTRACT FROM AUTHOR]

The Wiener–Kolmogorov theory of filtering has been with us since the first half of the twentieth century. A later matrix-based approach which was more general was derived with the steady-state Kalman filter. This approach uses a novel method of representing causal and uncausal systems in the form of convolution matrices and leads to a Wiener solution which is much easier to calculate than either the Kalman or Wiener approaches. For coloured additive noise, it avoids the use of Diophantine equations. The key idea missing in previous work is the close link between polynomials and Toeplitz matrices which are lower triangular in form. There is already a reasonably sized literature in the mathematics field on such matrices and so the area is ripe for exploration. Although the method does not offer a different or better solution, it shows a completely new way of defining linear time-invariant (LTI) systems which is neither transfer-function nor state-space-based. This is achieved by exploiting the connection between polynomials and Toeplitz matrices. The application here is the Wiener filter but there could well be many more as this is a generic approach. [ABSTRACT FROM PUBLISHER]

Ultrasonic phased array systems are becoming increasingly popular as tools for the inspection of safety-critical structures within the non-destructive evaluation industry. The data-sets captured by these arrays can be used to image the internal structure of individual components, allowing the location and nature of any defects to be deduced. Although there exist strict procedures for measuring defects via these imaging algorithms, sizing flaws which are smaller than two wavelengths in diameter can prove problematic and the choice of threshold at which the defect measurements are made can introduce an aspect of subjectivity. This paper puts forward a completely objective approach specific to cracks based on the Kirchhoff scattering model and the approximation of the resulting scattering matrices by Toeplitz matrices. A mathematical expression relating the crack size to the maximum eigenvalue of the associated scattering matrix is derived. Analysis of this approximation shows that the method will provide a unique crack size for a given maximum eigenvalue whilst providing a quick calculation method which avoids the need to numerically generate model scattering matrices (the computation time is up totimes faster). A sensitivity analysis demonstrates that the method is most effective for sizing defects that are commensurate with or smaller than the wavelength of the ultrasonic wave. The method is applied to simulated FMC data arising from finite element calculations where the crack length to wavelength ratios range between 0.6 and 1.9. The recovered objective crack size exhibits an error of 12%. [ABSTRACT FROM PUBLISHER]

We prove that Toeplitz matrices are unitarily similar to complex symmetric matrices. Moreover, twounitary matrices that uniformly turn allToeplitz matrices via similarity to complex symmetric matrices are explicitly given, respectively. When, we prove that each complex symmetric matrix is unitarily similar to some Toeplitz matrix, but the statement is false when. [ABSTRACT FROM PUBLISHER]

We characterize continuous, nondegenerate multiplicative maps on the algebraofupper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of(for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily invariant norms, such isometries are precisely the unitary similarities, or unitary similarities composed with complex conjugation. We partially extend these results to algebras generated by non-derogatory matrices. [ABSTRACT FROM PUBLISHER]

In this paper we consider the solvability of the matrix equations,with respect to the semi-tensor product. Using the concept of the semi-tensor product which was initially proposed by Cheng et al. [An introduction to Semi-Tensor product of matrices and its applications, World Scientific, 2012], firstly we investigate the matrix–vector equations,under the semi-tensor product, establish some compatible conditions for the matrices, and propose a necessary and sufficient condition for the solvability of the matrix–vector equation. We then explore the solvability of the matrix equations,under the semi-tensor product. Some elementary examples are given to show the efficiency of the results. [ABSTRACT FROM AUTHOR]

Letandwe first solve the minimum Frobenius norm residual problem (Problem LSP):with unknown Toeplitz matricesXandY. We then consider a best approximation problem: given Toeplitz matricesand, findsuch thatwhereis the solution set of Problem LSP. We show that the best approximation solutionis unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

LetMbe an arbitrary complex manifold and letLbe a Hermitian holomorphic line bundle overM. We introduce the Berezin–Toeplitz quantization of the open set ofMwhere the curvature onLis nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k−N], whereN>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powersLk. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limitk→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles. [ABSTRACT FROM PUBLISHER]

In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Haoet al.[A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant (CS) representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Haoet al.[A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM PUBLISHER]

We study the asymptotic behaviour of the singular values of matrices with entriesifj|iand zero otherwise, withfan arithmetical function. In particular, we study the case wherefis multiplicative andis regularly varying. Our main result is that, under quite general conditions, the singular values are, asymptotically,, whereare the eigenvalues of some positive Hilbert–Schmidt operator. [ABSTRACT FROM AUTHOR]

In this paper, we study the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma. At first, we study “complete convergence” versions of the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma. Two counterexamples show that they can fail in general and some sufficient conditions for “complete convergence” version of the Cesàro mean convergence theorem are given. Second, we introduce two classes of complete moment convergence, which are stronger versions of mean convergence and consider the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma under these two classes of complete moment convergence. [ABSTRACT FROM PUBLISHER]

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms. [ABSTRACT FROM PUBLISHER]

In the analysis of stationary stochastic process, one has to deal with covariance matrix of Toeplitz (or Laurent) structure. Such structure has a feature that not only the elements on the principal diagonal but also those lying on each of the parallel sub-diagonals are equal as well. The present investigation is on the problem of large sample testing of the Toeplitz pattern of the population covariance matrix. Apart from usual application of likelihood ratio and Rao's efficient score criteria, some heuristic two-stage tests are suggested. The results ofMonte Carlo experiment are reported for the size of the proposed tests. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of Toeplitz matrices with respect to different matrix products. We also give some results regarding circulant matrices, skew-circulant matrices and approximation by Toeplitz matrices over the field of complex numbers. [ABSTRACT FROM AUTHOR]

Iterative methods such as generalized minimal residual (GMRES) method are used to solve large sparse linear systems. This paper is considered the GMRES method for solvingtridiagonal block Toeplitz linear systemswithdiagonal blocks, and establishes upper bounds for GMRES residuals. The coefficient matrixbecomes an m-tridiagonal Toeplitz matrix, and tridiagonal toeplitz systems are subcases of these systems. Also, we show that the GMRES method onlinear systemcomputes the exact solution in at mostNsteps. [ABSTRACT FROM PUBLISHER]

One of the most serious problems in relay systems is power constraint. Due to this limitation, complex algorithms are replaced by cost efficient techniques. In conventional orthogonal frequency division multiplexing (OFDM) systems at the relays, signal processing techniques, such as equalization is performed considering signalling in the frequency domain. Employing such techniques, the received OFDM time signals in the first hop have to be converted into frequency domain at the relays, equalized and converted back to time domain again via the next hop for retransmission. In this paper, by employing the Toeplitz matrix structure that converts frequency selective channels into an equivalent channel matrix, the convolution of input signal and the channel is transferred into multiplication of input signal with the equivalent channel matrix causing the removal of the OFDM procedure at the relay. Since the span of the channel is calculated by a channel matrix, the signal length is not increased causing the transmission rate to be increased without the loss of full system BER. [ABSTRACT FROM PUBLISHER]

We introduce a new sparse estimator of the covariance matrix for high-dimensional models in which the variables have a known ordering. Our estimator, which is the solution to a convex optimization problem, is equivalently expressed as an estimator that tapers the sample covariance matrix by a Toeplitz, sparsely banded, data-adaptive matrix. As a result of this adaptivity, the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators. In particular, our convex banding estimator is minimax rate adaptive in Frobenius and operator norms, up to log factors, over commonly studied classes of covariance matrices, and over more general classes. Furthermore, it correctly recovers the bandwidth when the true covariance is exactly banded. Our convex formulation admits a simple and efficient algorithm. Empirical studies demonstrate its practical effectiveness and illustrate that our exactly banded estimator works well even when the true covariance matrix is only close to a banded matrix, confirming our theoretical results. Our method compares favorably with all existing methods, in terms of accuracy and speed. We illustrate the practical merits of the convex banding estimator by showing that it can be used to improve the performance of discriminant analysis for classifying sound recordings. Supplementary materials for this article are available online. [ABSTRACT FROM PUBLISHER]

The generalized minimal residual (GMRES) method is widely used to solve a linear system. This paper establishes upper and lower bounds for GMRES residuals for solving antridiagonal Toeplitz linear system. For normal matrixA, this problem has been studied previously by Li [Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47(3) (2007), 577–599.]. Also, Li and Zhang [The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer. Math. 112 (2009), pp. 267–293.] for non-symmetric matrixA, presented upper bound for GMRES residuals. In fact, our main goal in this paper is to find the upper and lower bounds for GMRES residuals on normal tridiagonal Toeplitz linear systems, and lower bounds for residuals of GMRES on solving non-normal tridiagonal Toeplitz linear systems. [ABSTRACT FROM PUBLISHER]

The objective of this paper is to obtain an upper bound to theHankel determinant for certain subclass ofp-valent functions, using Toeplitz determinants. [ABSTRACT FROM PUBLISHER]

Letbe the set of Toeplitzmatrices. We describe the matricesin the unitary groupsuch that [ABSTRACT FROM AUTHOR]

In this paper, we characterize idempotent Toeplitz sign patterns and idempotent Hankel sign patterns. Our work extends some existing results. [ABSTRACT FROM AUTHOR]

A Hankel circulant is a matrix obtained by reversing the order of columns (or rows) in a conventional circulant. A Toeplitz-plus-Hankel circulant (briefly, ()-circulant) is the sum of a circulant and a Hankel circulant. Bozzo discovered that the setof ()-circulants is the centralizer of the matrix, whereis the cyclic permutation matrix. As a consequence, all the matrices incan be simultaneously brought to a block diagonal form with diagonal blocks of orders one and two by a unitary similarity transformation. We show that the same assertion holds forif unitary similarities are replaced by unitary congruences. [ABSTRACT FROM AUTHOR]

The article focuses on consequences of Szegö's distribution theorem. Topics discussed include formation of Toeplitz matrices, how different eigenvalues of convex combination could approximate to make the absolute error tending to be zero and pertinence of Lebesgue integral function and Riemann integral numbers in the study.

An implicit second-order finite difference scheme, which is unconditionally stable, is employed to discretize fractional advection–diffusion equations with constant coefficients. The resulting systems are full, unsymmetric, and possess Toeplitz structure. Circulant and skew-circulant splitting iteration is employed for solving the Toeplitz system. The method is proved to be convergent unconditionally to the solution of the linear system. Numerical examples show that the convergence rate of the method is fast. [ABSTRACT FROM PUBLISHER]

Motivated by the work of Nazarov and Shapiro on the unit disk, we study asymptotic Toeplitzness of composition operators on the Hardy space of the unit sphere in. We extend some of their results but we also show that new phenomena appear in higher dimensions. [ABSTRACT FROM PUBLISHER]