Showing papers on "Discrete orthogonal polynomials published in 1986"
TL;DR: In this paper, the authors show that the convergence and absolute convergence of orthogonal polynomials on infinite intervals and on the untt crrcle can be explained by the convergence of Christoffel functions.
372 citations
TL;DR: A new method of converting between the direct form predictor coefficients and line spectral frequencies is presented, which is highly accurate and can be used in a form that avoids the storage of trigonometric tables or the computation of trig onometric functions.
Abstract: Line spectral frequencies provide an alternate parameterization of the analysis and synthesis filters used in linear predictive coding (LPC) of speech. In this paper, a new method of converting between the direct form predictor coefficients and line spectral frequencies is presented. The system polynomial for the analysis filter is converted to two even-order symmetric polynomial with interlacing roots on the unit circle. The line spectral frequencies are given by the positions of the roots of these two auxiliary polynomials. The response of each of these polynomials on the unit circle is expressed as a series expansion in Chebyshev polynomials. The line spectral frequencies are found using an iterative root finding algorithm which searches for real roots of a real function. The algorithm developed is simple in structure and is designed to constrain the maximum number of evaluations of the series expansions. The method is highly accurate and can be used in a form that avoids the storage of trigonometric tables or the computation of trigonometric functions. The reconversion of line spectral frequencies to predictor coefficients uses an efficient algorithm derived by expressing the root factors as an expansion in Chebyshev polynomials.
257 citations
TL;DR: In this paper, the limit distribution of polynomials is characterized by one (for each ) extremal relation with a variable (depending on ) weight function, and the authors prove a theorem which characterizes the limit distributions of the zeros.
Abstract: The authors prove a theorem which characterizes the limit distribution of the zeros of polynomials , , defined by one (for each ) extremal relation with a variable (depending on ) weight function.Bibliography: 9 titles.
189 citations
TL;DR: In this article, the authors investigate the properties of polynomials whose coefficients are asymptotically periodic and construct the measure with respect to which the polynomial coefficients are orthogonal and discuss their asymptic behavior.
117 citations
01 Jan 1986
TL;DR: In this paper, a Favard type theorem for special sequences of Laurent polynomials is proved, which is used to establish the relation between T -fractions and orthogonal Laurent poynomials (OLPs).
Abstract: A Favard type theorem for special sequences of Laurent polynomials is proved. This result is used to establish the relation between T -fractions and orthogonal Laurent polynomials (OLPs). Applications are given to T -fractions for (quotients of) hypergeometric functions of type 2 F 1 and confluent forms. In the special case of 2 F 1 ( a ,1; c ; z ) a weight function is given on the unit circle in C for the corresponding OLPs.
96 citations
TL;DR: In this paper, the coefficients of the three term recurrence formula satisfied by the orthogonal polynomials were investigated in terms of the coefficients this paper for measures associated with orthogonality.
Abstract: Properties of measures associated with orthogonal polynomials are investigated in terms of the coefficients of the three term recurrence formula satisfied by the orthogonal polynomials.
85 citations
01 Jan 1986
TL;DR: In this article, a CLT for processes of the form L(Xt) is proved, where L(x) is a polynomial and Xt, t ∈ ℤ is a process with long range dependence.
Abstract: A CLT for processes of the form L(Xt) is proved, where L(x) is a polynomial and Xt, t ∈ ℤ is a process with long range dependence. Conditions on Xt are formulated in terms of semi-invariants; they are specified for linear processes Xt. The notion of the Appell rank of L(x) plays a basic role in the CLT. Various topics related to Appell polynomials (e.g. expansions, diagram formalism for semi-invariants) are discussed.
80 citations
TL;DR: In this paper, an algorithm of polynomial complexity is described for factoring polynomials in several variables into irreducible factors over a field F which is finitely generated over the prime subfield H.
Abstract: An algorithm of polynomial complexity is described for factoring polynomials in several variables into irreducible factors over a field F which is finitely generated over the prime subfield H. An algorithm is also constructed for finding the components of the protective variety of common roots of homogeneous polynomials
(let c−1 denote its dimension) with working time polynomial in
. where
, the number L is the size of the representation of the polynomials
and
.
73 citations
TL;DR: In this paper, the Schur-Cohn stability test for polynomials with complex coefficients was extended to polynomial with complex polynomorphisms, which is based on a three-term recursion of a conjugate symmetric sequence of polynoms.
65 citations
TL;DR: In this paper, it was shown that for any automorphism φ of C[x, y], the face polynomials are the face automorphisms of the automorph group of C [x 1,…,xn] for n>2.
59 citations
TL;DR: In this article, necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials corresponding to measures dα with support [−1, 1] for which α′ > 0 almost everywhere in [− 1, 1].
TL;DR: In this article, the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the Jacoby and the Gegenbauer poynomials are studied.
TL;DR: In this article generalized Laguerre polynomials are defined and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix.
Abstract: Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.
01 Jan 1986
TL;DR: In this paper, the authors discuss random polynomials other than algebraic, that is, random trigonometric polynoms, random orthogonal poynomials, and random hyperbolic polynomorphisms.
Abstract: This chapter discusses random polynomials other than algebraic, that is, random trigonometric polynomials, random orthogonal polynomials, and random hyperbolic polynomials. It explains the relationship between random algebraic polynomials and other types of random polynomials. Any random orthogonal polynomial can be written as a random algebraic polynomial using known polynomial representations of orthogonal polynomials. The chapter also describes the number and expected number of real zeros of random trigonometric polynomials. It then discusses random hyperbolic polynomials and the expected number of real zeros of these polynomials. The chapter discusses the average number of real zeros of random orthogonal polynomials. It presents some numerical results on the number of real zeros of the random polynomials considered in the chapter and some figures that illustrate the distribution of the real zeros of the random polynomials considered here.
01 Dec 1986
TL;DR: In this article, sufficient conditions for the robust stability of Schur polynomial as a function of a single parameter are derived. But these conditions are not applicable to the robustness of Hurwitz and Schur under parameter variations.
Abstract: In this paper sufficient conditions for the robust stability of the Schur polynomial as a function of a single parameter are obtained. The conditions derived are based on a recent paper by Bose and Zeheb [1]. Also, based on the recent results of Nie and Xie [2] and Lipatov and Sokolov [9] simple sufficient conditions for the robust stability of both (strictly) Hurwitz and Schur polynomials as functions of a parameter are obtained. Furthermore, sufficient conditions for robustness of (strictly) Hurwitz and Schur polynomials under parameter variations are derived. Several examples are discussed illustrating the derived results. The derived results of this paper mainly extend the works of Kharitonov [3] and others.
TL;DR: A survey of invariant polynomials with matrix arguments and their applications in multivariate distribution theory including related developments in econometrics can be found in this article, where the authors survey the properties of invariants and their application in multi-dimensional distribution theory.
Abstract: Invariant polynomials with matrix arguments have been defined by the theory of group representations, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interest in the polynomials has been shown by people working in the field of econometric theory. In this paper, we shall survey the properties of the invariant polynomials and their applications in multivariate distribution theory including related developments in econometrics.
TL;DR: In this paper, the operational matrices for forward or backward integration of general orthogonal polynomials are derived and the tensor expression for the generalized orthogonality approximation of any two arbitrary functions is also introduced.
Abstract: The operational matrices for forward or backward integration of general orthogonal polynomials are derived The tensor expression for the generalized orthogonal polynomial approximation of any two arbitrary functions is also introduced It is shown that the approximation solution obtained using the Chebyshev polynomial is readily obtainable as a special case of the results derived Thus, the present results include results presented by Shih in 1983 and Chou and Horng in 1984 A linear time-varying optimal-control system with a quadratic performance measure is solved by using the generalized orthogonal polynomials Comparison of the results with those obtained using several different classical orthogonal polynomial approximations is also included
TL;DR: A q -analogue of Palama's limit, obtaining Hermite polynomials from Laguerre polynomial as the parameter α → ∞, is given in this article, and the corresponding limit for a pair of weight functions is obtained.
TL;DR: On donne une version discrete de la propriete de quasi-orthogonalite appliquee a une extension de la classe des polynomes de Meixner as discussed by the authors.
TL;DR: A Conjectured Analogue of Rolle's Theorem for Polynomials with Real or Complex Coefficients as discussed by the authors is presented in the American Mathematical Monthly: Vol 93, No. 1, pp. 8-13.
Abstract: (1986). A Conjectured Analogue of Rolle's Theorem for Polynomials with Real or Complex Coefficients. The American Mathematical Monthly: Vol. 93, No. 1, pp. 8-13.
TL;DR: Asymptotics for the greatest zeros of symmetric orthogonal polynomials are investigated in this paper in terms of the asymptotic behavior of the recursion coefficients.
Abstract: Asymptotics for the greatest zeros of symmetric orthogonal polynomials is investigated in terms of the asymptotic behavior of the recursion coefficients.
TL;DR: In this article, a method of constructing orthogonal polynomials generated by pairs of Hermitian operators in representations of Lie algebras is presented, where the Hermitians are replaced by a set of Lie algebraic operators.
Abstract: We present a method of constructing orthogonal polynomials generated by pairs of Hermitian operators in representations of Lie algebras. All known classical polynomials of both discrete and continuous argument are generated naturally by the simplest Lie algebras.
TL;DR: In this article, the authors derived explicit representations and complete asymptotic expansions for the Askey-Wilson polynomials and the little and big q-Jacobi poynomials.
Abstract: We derive explicit representations and complete asymptotic expansions for the Askey–Wilson ${}_4 \phi _3 $ polynomials and the little and big q-Jacobi polynomials. We also give an alternate proof of a Dirichlet–Mehler type formula for the continuous q-ultraspherical polynomials. We also determine the asymptotic behavior of the q-Racah polynomials.
TL;DR: In this paper, the Kleiser-Schumann algorithm for the approximation of the Stokes problem by Fourier/Legendre polynomials is analized, and stability when the degree of the polynomial increases is established, whereas error estimates in Sobolev spaces are proven.
Abstract: The Kleiser-Schumann algorithm for the approximation of the Stokes problem by Fourier/Legendre polynomials is analized. Stability when the degree of the polynomials increases is established, whereas error estimates in Sobolev spaces are proven.
TL;DR: In this article, the generalized orthogonal polynomials (GOP) is applied to the analysis and optimal control of time-varying systems, and the operational matrix for the forward and backward integration of the generalized Orthogonal Polynomial (OGP) is derived.
Abstract: The method or generalized orthogonal polynomials (GOP) is applied to the analysis and optimal control of time-varying systems. The proposed GOP can represent all kinds of individual orthogonal-polynomial and non-orthogonal Taylor series. The operational matrix for the forward and backward integration of the generalized orthogonal polynomials, and the operational matrix of the product of r' and the generalized orthogonal-polynomial vector are derived and applied to time-varying systems. By using these three kinds of operational matrices, the computational algorithm for calculating the expansion coefficients is very simple and effective. Three satisfactory examples illustrate the usefulness of the method.
Journal Article•
TL;DR: In this paper, the authors define multivariate Meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms.
TL;DR: In this paper, the shift transformation matrix of general discrete orthogonal polynomials is introduced to simplify the discrete Euler-Lagrange equations into linear algebraic ones for the approximation of state and control variables of digital systems.
Abstract: The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.