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Summary: Gramian 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. [ABSTRACT FROM AUTHOR]

In this work, we provide theoretical estimates for the ranks of the power functions , in the quantized tensor train (QTT) format for . Such functions and their several generalizations (e.g., ) play an important role in studies of the asymptotic solutions of the aggregation-fragmentation kinetic equations. In order to support the constructed theory we verify the values of QTT-ranks of these functions in practice with the use of the TTSVD procedure and show an agreement between the numerical and analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative approach for assessing the stability of linear time-invariant (LTI) systems. The key innovations in our methodology include the calculation of Markov parameters, the extension to nonlinear continuous-time systems, and significantly improved computational efficiency. Our novel approach leverages Markov parameters derived from the characteristic polynomial of the system. Unlike traditional methods, we employ the Leverrier-Takeno algorithm to efficiently determine the coefficients of the characteristic polynomial for state-space representation. The systematic calculation of Markov parameters using a specialized Maclaurin series expansion enhances our method's precision. One of our major achievements is the rigorous verification of stability through Hankel matrices, ensuring positive definiteness using the Cholesky decomposition algorithm. In addition, we introduce a second stability criterion based on iterated polynomial long divisions, although we acknowledge its limitations in handling high-order systems due to computational inefficiency. Our pioneering contributions extend beyond LTI systems. We adapt our innovative techniques to analyze the local stability of nonlinear continuous-time systems by introducing the concept of Jacobian matrices and employing the indirect Lyapunov method. This expansion broadens the applicability of our approach to a wider range of real-world systems. Notably, our approach exhibits remarkable computational efficiency. According to CPU-time analysis, our first stability criterion outperforms the classical Routh–Hurwitz method by more than 12 times for dynamic systems with orders ranging from 2 to 180. This significant reduction in computation time positions our method as a valuable option for real-time stability analysis applications. [ABSTRACT FROM AUTHOR]

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We prove an exact formula for the variance of the divisor function over short intervals in A : = F q [ T ] , where q is a prime power; and for correlations of the form d (A) d (A + B) , where we average both A and B over certain intervals in A . We also obtain an exact formula for correlations of the form d (K Q + N) d (N) , where Q is prime and K and N are averaged over certain intervals with deg N ≤ deg Q - 1 ≤ deg K ; and we demonstrate that d (K Q + N) and d(N) are uncorrelated. We generalize our results to σ z defined by σ z (A) : = ∑ E ∣ A | A | z for all monics A ∈ A . Our approach is to use the orthogonality relations of additive characters on F q to translate the problems to ones involving the ranks of Hankel matrices over F q . 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 d k . [ABSTRACT FROM AUTHOR]

For B ∈ F q [ T ] of degree 2 n ≥ 2 , consider the number of ways of writing B = E 2 + γ F 2 , where γ ∈ F q ⁎ is fixed, and E , F ∈ F q [ T ] with deg E = n and deg F = m < n. We denote this restricted sum-of-squares function by S γ ; m (B). We obtain an exact formula for the variance of S γ ; m (B) over short intervals in F q [ T ] where q is an odd prime power. Interestingly, when m + 1 ≤ n ≤ 2 m , the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over F q , that are defined from Hankel matrices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We deal with absolutely continuous probability distributions with finite all-positive integer-order moments. It is well known that any such distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). In this paper, we follow the maximum entropy approach and establish a new criterion for the M-indeterminacy of distributions on the positive half-line (Stieltjes case). Useful corollaries are derived for M-indeterminate distributions on the whole real line (Hamburger case). We show how the maximum entropy is related to the symmetry property and the M-indeterminacy. [ABSTRACT FROM AUTHOR]

The smallest eigenvalue of large Hankel matrices generated by a semiclassical Laguerre weight, zαe−z²+tz, where z ∈ [0,∞), α > −1, and t ∈ R, can be obtained through the asymptotics of the orthonormal polynomials Pn(z) with respect to this weight. [ABSTRACT FROM AUTHOR]

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.

In this paper, we study the asymptotic behavior of the smallest eigenvalue \lambda _N, of the (N+1)\times (N+1) Hankel matrix \mathcal {M}_N=(\mu _{j+k})_{0\le j,k\le N} generated by the semi-classical Hermite weight w(z,t)=|z|^\lambda \exp \left (-z^2+tz\right), z, t \in \mathbb {R}, \lambda >-1. An asymptotic expression of the orthonormal polynomials \mathcal {P}_N(z) with the semi-classical Hermite weight w(z,t) is established as N tends to infinity. Based on the orthonormal polynomials \mathcal {P}_N(z), we obtain the specific asymptotic formulas of \lambda _{N}. [ABSTRACT FROM AUTHOR]

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger's results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition, we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) to exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices (these bounds are used for concluding indeterminacy); (c) to provide new arguments to confirm classical results; (d) to give new numerical illustrations involving commonly used probability distributions. [ABSTRACT FROM AUTHOR]

We deal with absolutely continuous probability distributions with finite all-positive integer-order moments. It is well known that any such distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). In this paper, we follow the maximum entropy approach and establish a new criterion for the M-indeterminacy of distributions on the positive half-line (Stieltjes case). Useful corollaries are derived for M-indeterminate distributions on the whole real line (Hamburger case). We show how the maximum entropy is related to the symmetry property and the M-indeterminacy.

Let n be a positive Borel measure on the interval [0;¥). Let Hv =(Vn,k)n,k≥0 be the Hankel matrix with entries Vn,k = F[0;¥) tn+k/n! dv(t). The matrix Hn induces formally the operator Hv (f)(z)=...k=0 nn;kak)zn on the space of all entire functions f(z)=... In this paper, we investigate those positive Borel measures such thatHn (f...Fp, and among them we characterize those for which Hn is a bounded (resp., compact) operator from the Fock space Fq into the space Fq (...). [ABSTRACT FROM AUTHOR]

We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums. [ABSTRACT FROM AUTHOR]

We study the relation between certain non-degenerate lower Hessenberg infinite matrices G and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for Sobolev orthogonality. We characterize the structure of the matrix G and the associated matrix of formal moments M G in terms of certain matrix operators. [ABSTRACT FROM AUTHOR]

The Hankel matrices considered in this article arose in one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that, in the case of the Riemann zeta function, the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when the zeta function is replaced by a function having a single zero. For this case, we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors. [ABSTRACT FROM AUTHOR]

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials, and Narayana polynomials. [ABSTRACT FROM AUTHOR]

Abstract In skywave radar, the slow‐time samples received in a certain range‐azimuth cell are usually processed for signal analysis and target detection. Particularly, to extract the principal components, such as sea clutter and target signal, in slow‐time samples, the Hankel matrix structure and subspace‐based methods are often applied, which can also reduce the additive Gaussian noise. In the real environment, the slow‐time samples sometimes are seriously contaminated by the transient interference, affecting the performance of useful signal reconstruction. The robust low‐rank matrix recovery methods have been applied to solve this problem, where the transient interference is assumed to be sparse, following the Laplace distribution. However, the Hankel structures of useful signal and transient interference have not been fully utilized in the conventional methods, which can enhance the reconstruction performance. Here, the robust low‐rank Hankel matrix recovery problems is reformulated for skywave radar slow‐time samples and solve them with the inexact augmented Lagrange multiplier method. Additionally, the low‐rank Hankel matrix completion problem is discussed, where the location of the transient interference is determined. The experimental results have demonstrated the good performance of our proposed methods and also some interesting conclusions are obtained.

In this paper, we study the large N behavior of the smallest eigenvalue λ N of the (N + 1) × (N + 1) Hankel matrix, H N = (μ j + k) 0 ≤ j , k ≤ N , generated by the γ dependent Jacobi weight w (z , γ) = e − γ z z α (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 P N (z) , z ∈ C ﹨ [ 0 , 1 ] , with the weight w (z , γ) = e − γ z z α (1 − z) β. Using the polynomials P N (z) , we obtain the theoretical expression of λ N , for large N. We also display the smallest eigenvalue λ N for sufficiently large N , computed numerically. [ABSTRACT FROM AUTHOR]

Abstract The nuclear magnetic resonance (NMR) spectroscopy has fruitful applications in chemistry, biology and life sciences, but suffers from long acquisition time. Non‐uniform sampling is a typical fast NMR method by undersampling the time‐domain data of the spectrum but need to restore the fully sampled data with proper constraints. The state‐of‐the‐art method is to model the time‐domain data as the sum of exponential functions and reconstruct these data by enforcing the low rankness of Hankel matrix. However, this method is solved by minimizing the sum of singular values of the Hankel matrix, which leads to the distortion of low‐intensity spectral peaks. Here, a low rank Hankel matrix reconstruction approach with a partial sum of singular values is proposed to protect small singular values, which can faithfully reconstruct all peaks. Results on both synthetic and realistic NMR spectroscopy show that the proposed method can reconstruct a more consistent spectrum to the fully sampled one than other state‐of‐the‐art methods and have particular advantages on preserving low‐intensity peaks.

An asymptotic expression for the polynomials Pn(z), z ∉ (−∞, ∞), orthonormal with respect to a singularly perturbed Gaussian weight, exp(−z2−t/z2), z ∈ (−∞,∞), t > 0, is established. Based on this, the asymptotic behavior of the smallest eigenvalue of the Hankel matrix generated by the weight is described. [ABSTRACT FROM AUTHOR]

Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling–Whitney–Riordan triangle [ T n , k ] n , k satisfying the recurrence relation: T n , k = (b 1 k + b 2) T n - 1 , k - 1 + [ (2 λ b 1 + a 1) k + a 2 + λ (b 1 + b 2) ] T n - 1 , k + λ (a 1 + λ b 1) (k + 1) T n - 1 , k + 1 , where initial conditions T n , k = 0 unless 0 ≤ k ≤ n and T 0 , 0 = 1 . We prove that the Stirling–Whitney–Riordan triangle [ T n , k ] n , k is x -totally positive with x = (a 1 , a 2 , b 1 , b 2 , λ) . We show that the row-generating function T n (q) has only real zeros and the Turán-type polynomial T n + 1 (q) T n - 1 (q) - T n 2 (q) is stable. We also present explicit formulae for T n , k and the exponential generating function of T n (q) and give a Jacobi continued fraction expansion for the ordinary generating function of T n (q) . Furthermore, we get the x -Stieltjes moment property and 3- x -log-convexity of T n (q) and show that the triangular convolution z n = ∑ i = 0 n T n , i x i y n - i preserves Stieltjes moment property of sequences. Finally, for the first column (T n , 0) n ≥ 0 , we derive some properties similar to those of (T n (q)) n ≥ 0. [ABSTRACT FROM AUTHOR]

Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A, 74 (1996), pp. 57--70]. Schirmacher [J. Combin. Theory Ser. A, 85 (1999), pp. 127--134] extended them to q-log-concavity and almost q-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula, and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle, and some triangles related to binomial coefficients, rook polynomials, and Laguerre polynomials. We not only get total positivity of these lower-triangles, and q-Stieltjes moment properties and 3-q-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve the Stieltjes moment property. In particular, we solve a conjecture of Sokal on the q-Stieltjes moment property of rook polynomials. [ABSTRACT FROM AUTHOR]

We compute the asymptotic distribution of the eigenvalues of a family of Hankel matrices. We also present some applications to centro-symmetric, skew-centro-symmetric and normal Hankel matrices and prove an extended version of a conjecture due to W. P. Angerer on the spectral distribution of band Hankel matrices. [ABSTRACT FROM AUTHOR]

The aim of this note is to provide an interpretation of two families of quasi anti-tridiagonal matrices in terms of graph theory. Understanding this junction allows us to recover a plethora of recent results on spectra, powers, inverse problems, and determinant, among others, of such matrices. [ABSTRACT FROM AUTHOR]

Any matrix A has an LU decomposition up to a row or column permutation. Less well-known is the fact that it has a 'Toeplitz decomposition' A = T 1 T 2 ⋯ T r where T i 's are Toeplitz matrices. We will prove that any continuous function f : R n → R m has an approximation to arbitrary accuracy by a neural network that maps x ∈ R n to L 1 σ 1 U 1 σ 2 L 2 σ 3 U 2 ⋯ L r σ 2 r − 1 U r x ∈ R m , i.e., where the weight matrices alternate between lower and upper triangular matrices, σ i (x) ≔ σ (x − b i) for some bias vector b i , and the activation σ may be chosen to be essentially any uniformly continuous nonpolynomial function. The same result also holds with Toeplitz matrices, i.e., f ≈ T 1 σ 1 T 2 σ 2 ⋯ σ r − 1 T r to arbitrary accuracy, and likewise for Hankel matrices. A consequence of our Toeplitz result is a fixed-width universal approximation theorem for convolutional neural networks, which so far have only arbitrary width versions. Since our results apply in particular to the case when f is a general neural network, we may regard them as LU and Toeplitz decompositions of a neural network. The practical implication of our results is that one may vastly reduce the number of weight parameters in a neural network without sacrificing its power of universal approximation. We will present several experiments on real data sets to show that imposing such structures on the weight matrices dramatically reduces the number of training parameters with almost no noticeable effect on test accuracy. [ABSTRACT FROM AUTHOR]

The accuracy of ultra wide band (UWB) scattering centre parameter estimation is the key bases in the study of target classification and identification. A novel 2D scattering centre parameter estimation algorithm for microwave chamber target is presented here. First, by using GTD model, this method constructs the microwave chamber measured data. Second, using matrix pencil method to build Hankel Matrix as a mathematic progress to estimate position parameter of scattering centres. Third, the type parameter and reflection intensity are estimated based on the least square method and principal components analysis. Using pre-estimated reflection intensity and clean technique to correctly determine scattering centre number and precisely estimate the reflection intensity of every scattering centres even if the intensity gap between the scattering centres is huge. The simulation results show that this novel approach can accurately and effectively determine the number and the parameters of the scattering centres. The Cramer–Rao lower bound (CRLB) is also given.

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials. [ABSTRACT FROM AUTHOR]

The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szegő theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real) nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix) are negative/positive for sufficiently large dimension, i.e. the matrix sequence is symmetric (asymptotically) indefinite. [ABSTRACT FROM AUTHOR]

Detection and measurement of partial discharge (PD) phenomena combined with the separation and identification of PD sources is the way to achieve effective insulation integrity assessment. However, during measurement, PD signals are coupled with interferences (discrete spectral, pulsive, and white noises). Recovering PD signals from such interferences would improve PD source separation (thus identification), but still remains a challenging task. Several denoising methods have been proposed to suppress interferences. However, using a universal method to achieve interference removal is probably impossible, as the characteristics of the interferences are distinct. This study proposes a novel low‐rank H‐Matrix‐based singular value decomposition (SVD) filter (H‐SVD) that removes different types of interferences. Denoising is done by projecting the measured pulse in a lower dimensional signal space. To assess the effectiveness of the proposed method, H‐SVD filter is first applied to simulated PD data and later on real‐time PD data with the introduction of three different types of synthetic noises. The results of the evaluation metrics confirm that H‐SVD has significant performance improvements compared to existing state‐of‐the‐art PD denoising methods. [ABSTRACT FROM AUTHOR]

The accuracy of ultra wide band (UWB) scattering centre parameter estimation is the key bases in the study of target classification and identification. A novel 2D scattering centre parameter estimation algorithm for microwave chamber target is presented here. First, by using GTD model, this method constructs the microwave chamber measured data. Second, using matrix pencil method to build Hankel Matrix as a mathematic progress to estimate position parameter of scattering centres. Third, the type parameter and reflection intensity are estimated based on the least square method and principal components analysis. Using pre-estimated reflection intensity and clean technique to correctly determine scattering centre number and precisely estimate the reflection intensity of every scattering centres even if the intensity gap between the scattering centres is huge. The simulation results show that this novel approach can accurately and effectively determine the number and the parameters of the scattering centres. The Cramer–Rao lower bound (CRLB) is also given. [ABSTRACT FROM AUTHOR]

Given a Lebesgue integrable function f over [-π; π], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as ±|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273{288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as π1 under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed. [ABSTRACT FROM AUTHOR]

Abstract We establish a connection between the local singular value decomposition and the geometry of n -dimensional curves. In particular, we link the left singular vectors to the Frenet-Serret frame, and the generalized curvatures to the singular values. Specifically, let γ : I → R n be a parametric curve of class C n + 1 , regular of order n. The Frenet-Serret apparatus of γ at γ (t) consists of a frame e 1 (t) , ... , e n (t) and generalized curvature values κ 1 (t) , ... , κ n − 1 (t). Associated with each point of γ there are also local singular vectors u 1 (t) , ... , u n (t) and local singular values σ 1 (t) , ... , σ n (t). This local information is obtained by considering a limit, as ϵ goes to zero, of covariance matrices defined along γ within an ϵ -ball centered at γ (t). We prove that for each t ∈ I , the Frenet-Serret frame and the local singular vectors agree at γ (t) and that the values of the curvature functions at t can be expressed as a fixed multiple of a ratio of local singular values at t. To establish this result we prove a general formula for the recursion relation of a certain class of sequences of Hankel determinants using the theory of monic orthogonal polynomials and moment sequences. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the smallest eigenvalue, λN, of the Hankel (or moments) matrix denoted by HN=μm+n0≤m,n≤N, with respect to the weight w(x)=xαe−xβ,x∈[0,∞),α>−1,β>12. An asymptotic expression of the polynomials orthogonal with w(x) is established. Using this, we obtain the specific asymptotic formulas of λN in this paper. Applying a parallel numerical algorithm, we get a variety of numerical results of λN corresponding to our theoretical calculations. [ABSTRACT FROM AUTHOR]

In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of { Y n T n (f) } n , where T n (f) ∈ R n × n is a real Toeplitz matrix generated by a function f ∈ L 1 ([ - π , π ]) , and Y n is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of Y n T n (f) are asymptotically described by a 2 × 2 matrix-valued function, whose eigenvalue functions are ± | f | . It turns out that roughly half of the eigenvalues of Y n T n (f) are well approximated by a uniform sampling of |f| over [ - π , π ] , while the remaining are well approximated by a uniform sampling of - | f | over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract The Stieltjes problem of moments seeks for a nondecreasing positive distribution function μ (λ) on the semi-axis [ 0 , + ∞) so that its moments match a given infinite sequence of positive real numbers m 0 , m 1 , .... In his seminal paper Investigations on continued fractions published in 1894 Stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal, the convergence theory of continued fractions. One can also reformulate the Stieltjes problem of moments as looking for a sequence of positive distribution functions μ (1) (λ) , μ (2) (λ) , ... , where the n th distribution function has n points of increase and m 0 , m 1 , ... , m 2 n − 1 represent its (first) 2 n moments, i.e., as the sequence of the finite Stieltjes moment problems. This view can be linked to iterative solution of (large) linear algebraic systems. Providing that m 0 , m 1 , ... are moments of some linear, self-adjoint and coercive operator A on a Hilbert space with respect to a given vector f , the finite Stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving A u = f , and vice versa. Here the existence and uniqueness is guaranteed by the properties of the operator A (reformulation for finite sequences, matrices and finite vectors is obvious). This fundamental link raises a question on how the solution of the finite Stieltjes moment problem can be described purely algebraically. This has motivated the presented exposition built upon ideas published previously by several authors. Since the description uses matrices of moments, it is not intended for numerical computations. [ABSTRACT FROM AUTHOR]