Displacement structure approach to q-adic Chebyshev–Vandermonde-like matrices

In this paper, a new class of so-called q-adic Chebyshev–Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the ordinary Chebyshev–Vandermonde-like matrices over the complex field studied earlier by Kailath and Olshevsky [T. Kailath, V. Olshevsky, Displacement structure approach to Chebyshev–Vandermonde and related matrices, Integral Equations Operator Theory 22 (1995) 65–92], and the classical q-adic Vandermonde-like matrices with respect to power basis by Yang and Hu [Z.H. Yang, Y.J. Hu, Displacement structure and fast inversion formulas for q-adic Vandermonde-like matrices, J. Comput. Appl. Math. 176 (2005) 1–14]. Three kinds of displacement structures and consequently, three kinds of fast inversion formulas are presented for this class of matrices by using displacement structure theory method, which generalize the corresponding results for Chebyshev–Vandermonde-like and q-adic Vandermonde-like matrices.     

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

Weighted pseudoinversion of matrices with singular weights

Weighted pseudoinverse matrices with singular weights are represented in terms of the coefficients of characteristic polynomials of certain square matrices. By using the expression obtained, we construct limit representations of weighted pseudoinverse matrices with singular weights.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1323–1327, October, 1994.     

The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Let μ be a compactly supported finite Borel measure in ℂ, and let Π_n be the space of holomorphic polynomials of degree at most n furnished with the norm of L ²(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Π_n their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞.Original Russian Text Copyright © 2005 Dovgoshei A. A., Abdullaev F., and Kucukaslan M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 774–785, July–August, 2005.     

Polynomials of Least Deviation from Zero

A new family of polynomials of least deviation from zero is defined on the unit disk B. Lower bounds for best approximations in the space L _p (B), p ≥ 1, are given.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 308–313.Original Russian Text Copyright © 2005 by V. A. Yudin.     

基于两类切比雪夫多项式的循环矩阵行列式的计算

利用多项式因式分解的逆变换,结合循环矩阵和切比雪夫多项式的特殊结构,首先研究第三类和第四类切比雪夫多项式的通项公式,并给出第三类、第四类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式的显式表达式,最后给出算法实施步骤.     

Computation of differentiating matrices for classical orthogonal polynomials of continuous and discrete argument

Recurrences are derived for computing the elements of the differentiating matrix for classical orthogonal polynomials of continuous and discrete argument. Two approaches to construction of difference differentiation formulas are considered. Examples of differentiating matrices for Hahn and Chebyshev orthogonal polynomials of discrete argument are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 19–24, 1992.     

Interlacing properties of roots of certain biorthogonal polynomials

In this paper we consider the biorthogonal polynomials with respect to the measure e^-x4-y2+2τxydxdy, and show that their roots interlace. The proof involves showing total nonnegativity of matrices related to Jacobi type matrices.     

On the Properties of Accretive-Dissipative Matrices

Let A be a complex n × n matrix, and let A = B + iC, B = B*, C = C* be its Toeplitz decomposition. Then A is said to be (strictly) accretive if B > 0 and (strictly) dissipative if C > 0. We study the properties of matrices that satisfy both these conditions, in other words, of accretive-dissipative matrices. In many respects, these matrices behave as numbers in the first quadrant of the complex plane. Some other properties are natural extensions of the corresponding properties of Hermitian positive-definite matrices.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 832–843.Original Russian Text Copyright ©2005 by A. George, Kh. D. Ikramov.     

Jacobi Matrix Pair and Dual Alternative <Emphasis Type="Italic">q</Emphasis>-Charlier Polynomials

By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 614–621, May, 2005.     

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n×n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over Fq of given degrees and for the number of n×n Hankel matrices over Fq of a given rank.     

Probabilistic aspects of Al-Salam--Chihara polynomials

We solve the connection coefficient problem between the Al-Salam--Chihara polynomials and the -Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam--Chihara polynomials, and compute determinants of related Hankel matrices.

     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly-nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.     

Concerning an Example of Paskiewich

We generalize the construction proposed by A. Paskiewich of an example of an orthonormal system which establishes the sharpness of the Men’shov-Rademacher theorem. The connection of his example with that of Men’shov is elucidated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 286–291.Original Russian Text Copyright © 2005 by A. P. Solodov.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.     

Computations with half-range Chebyshev polynomials

An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) [5]. Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev-Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix.     

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form

W(t)=t^αe^-te^Att^Bt^B*e^A*t,