System Of Orthogonal Polynomials Research Articles

Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation

Rigorous pointwise asymptotics are established for semiclassical soliton ensembles (SSEs) of the focusing nonlinear Schrodinger equation using techniques of asymptotic analysis of matrix Riemann- Hilbert problems. The accumulation of poles in the eigenfunction is handled using a new method in which the residues are simultaneously interpolated at the poles by two distinct interpolants. The results justify the WKB approximation for the nonselfadjoint Zakharov-Shabat operator with real-analytic, bell-shaped, even potentials. The new technique introduced in this paper is applicable to other problems as well: (i) it can be used to provide a unified treatment by Riemann-Hilbert methods of the zero-dispersion limit of the Korteweg-de Vries equation with negative (soliton generating) initial as studied by Lax, Levermore, and Venakides, and (ii) it allows one to compute rigorous strong asymptotics for systems of discrete orthogonal polynomials. Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert problem amounts to reconstructing a sectionally meromorphic matrix from given homogeneous multiplicative at the boundary contours of the domains of meromorphy, from part data given at the prescribed singularities, and from a normalization condition. So, many asymptotic questions in integrable systems (e.g. long time behavior and singular perturbation theory) and approximation theory (e.g. behavior of orthogonal polynomials in the limit of large degree) amount to determining asymptotic properties of the solution matrix of a Riemann-Hilbert problem from given asymptotics of the jump conditions and principal part data. In recent years a collection of techniques has emerged for studying certain asymptotic problems of this sort. These techniques are analogous to familiar asymptotic methods for expanding oscillatory integrals, and we often refer to them as steepest-descent methods. The basic method first appeared in the work of Deift and Zhou (DZ93). The first applications were to Riemann-Hilbert problems without poles, in which the solution matrix is sectionally holomorphic. Later, some problems were studied in which there were a number of poles — a number held fixed in the limit of interest — in the solution matrix (see, for example, the paper (DKKZ96) on the long-time behavior of the Toda lattice with rarefaction initial data). The previous methods were extended to these more complicated problems through the device of making a local change of variable near each pole in some small domain containing the pole. The change of variable is chosen so that it has the effect of removing the pole at the cost of introducing an explicit jump on the boundary of the domain around the pole in which the transformation is made. The result is a Riemann-Hilbert problem for a sectionally holomorphic matrix, which can be solved asymptotically by pre-existing steepest-descent methods. Recovery of an approximation for the original sectionally meromorphic matrix unknown involves putting back the poles by reversing the explicit change of variables that was designed to get rid of them to begin with. Yet another category of Riemann-Hilbert problems consists of those problems where the number of poles is not fixed, but becomes large in the limit of interest, with the poles accumulating on some closed set F in the finite complex plane. A problem of this sort has been addressed (KMM00) by making an explicit transformation of the type described above in a single fixed domain G that contains the locus of accumulation F of all the poles. The transformation is chosen to get rid of all the poles at once. In order to specify it,

Realization of the Annihilation Operator for an Oscillator-Like System by a Differential Operator and Hermite-Chihara Polynomials

We obtain the differential operator realization for the annihilation operator A of generalized Heisenberg algebra corresponding to the given polynomial system. The important special case of orthogonal polynomial systems, for which the matrix of the operator A in l 2 ( Z + ) has only off-diagonal elements on the first upper diagonal different from zero, is considered. The known generalized Hermite polynomials give us an example of such an orthonormal system. The suggested approach can be applied to a similar investigation of various "deformed" polynomial systems.

Some Orthogonal Polynomials Related to Elliptic Functions

We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.

Transmutation of orthogonal polynomials*

We investigate the existence of transformation operators mapping systems of orthogonal polynomials generated by self-adjoint operators with different spectra. An interpolation between the spectra allows us to write a transmutation relation between functions of both operators. After showing a factorization result, we shall examine an example using hypergeometric functions

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}∞n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}∞n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1ℓi(x)y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}∞n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q n (x)} n=0 ∞, define another polynomial system by $P_{n}(x)=Q_{n}(x)-\alpha_{n}Q_{n-t}(x),~~ n\geq 0, $ where α n are complex numbers and t is a positive integer. We find conditions for {P n (x)} n=0 ∞ to be an orthogonal polynomial system. When t=1 and α1≠0, it turns out that {Q n (x)} n=0 ∞ must be kernel polynomials for {P n (x)} n=0 ∞ for which we study, in detail, the location of zeros and semi-classical character.

Characterization by orthogonal polynomial systems of finite Markov chains

The paper characterizes matriceswhich have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constrainta0j= 0 forj≥ 2 are specified. The only stochastic matricesP ={pij} satisfyingp00+p01= 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

Characterization by orthogonal polynomial systems of finite Markov chains

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

Chebychev polynomials in a speech recognition model

Advanced speech information processing systems require further research on speaker-dependent information. Recently, a specific system of discrete orthogonal polynomials {φ L r ( l); l = 1, 2, …, L} R r =0 has been encountered to play a dominant role in a segmental probability model recently proposed in the speaker-dependent feature extraction from speech waves and applied to text-independent speaker verification. Here, these speech polynomials are shown to be the shifted Chebyshev polynomials on a discrete variable t r ( l − 1, L), whose structural and spectral properties are discussed and reviewed in light of the recent discoveries in the field of discrete orthogonal polynomials.

Generalized Hahn's theorem

Let { P n ( x)} n=0 ∞ be an orthogonal polynomial system and L[·]= ∑ i=0 k a i(x) D i D= d dx a linear differential operator of order k (⩾0) with polynomial coefficients. We find necessary and sufficient conditions for a polynomial sequence { Q n ( x)} n=0 ∞ defined by Q n(x) ≔ L[P n+r (r)(x)],n⩾0 , to be also an orthogonal polynomial system. We also give a few applications of this result together with the complete analysis of the cases: (i) k=0,1,2 and r=0, and (ii) k= r=1.

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system $$p = \left( {p_n } \right)_{n \in N_0 } $$ and a sequence $$d = \left( {d_n } \right)_{n \in N} 0$$ of nonzero numbers,let $$S_{p,d} $$ be the linear operator defined on the linear spaceof all polynomials via $$S_{p,d} p_n = d_n p_n $$ for all $$n \in N_0 $$ .We investigate conditions on $$p$$ and $$d$$ under which $$S_{p,d} $$ can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for $$p = \left( {L_n^\alpha } \right)$$ , a generalized Laguerre polynomial system, no $$d$$ can simultaneously preserve the orthogonality of twoadditional Laguerre systems, $$\left( {L_n^{\alpha + t_1 } } \right)$$ and $$\left( {L_n^{\alpha + t_2 } } \right)$$ , where $$t_1 ,t_2 \ne 0$$ and $$t_1 \ne t_2 $$ . On the other hand, for $$p = \left( {T_n } \right)$$ ,the Chebyshev polynomial system and $$d = \left( {\left( { - 1} \right)^n } \right)$$ , $$S_{p,d} $$ simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.

Quasi‐definiteness of generalized Uvarov transforms of moment functionals

When σ is a quasi‐definite moment functional with the monic orthogonal polynomial system , we consider a point masses perturbation τ of σ given by , where λ, ulk, and cl are constants with ci ≠ cj for i ≠ j. That is, τ is a generalized Uvarov transform of σ satisfying A(x) τ = A(x) σ, where . We find necessary and sufficient conditions for τ to be quasi‐definite. We also discuss various properties of monic orthogonal polynomial system relative to τ including two examples.

The Lanczos algorithm for extensive many-body systems in the thermodynamic limit

We establish rigorously the scaling properties of the Lanczos process applied to an arbitrary extensive many-body system which is carried to convergence n→∞ and the thermodynamic limit N→∞ taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating function. The measure and the orthogonal polynomial system associated with the Lanczos process in this regime are also given explicitly. Some important representations of these Lanczos functions are given, including Taylor series expansions, and theorems controling their general properties are proven.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Asymptotic normality of the coefficients of polynomials related to the classical system orthogonal ones

Suitable transformations on the classical system of orthogonal polynomials lead to polynomials with nonnegative coefficients. Asymptotic normality of those coefficients is established by local limit theorem. Applications include, important combinatorial numbers, as the number of connections of a telephone exchange, the coefficients of a rook polynomial, the coefficients of the card matching polynomial and others. Asymptotic formulas for such numbers are also provided whenever it is possible.

Gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions

The theory of strong moment problems has provided Gaussian quadrature rules for approximate integration with respect to strong distributions. In Hagler (Ph.D. Thesis, University of Colorado, Boulder, 1997) and Hagler et al. (Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, in press), a transformation of the form v( x)=(1/ λ)( x− γ/ x), λ, γ>0, is used to obtain strong mass distribution functions from mass distribution functions. This transformation also links the systems of orthogonal polynomials and Laurent polynomials and their zeros. In this paper we show how the transformation method can be used to obtain the Gaussian quadrature rules for strong extensions of mass distribution functions. We then provide numerical examples of strong Gaussian quadrature approximations to the integrals of elementary functions with respect to selected strong distributions.

Spherical functions on the Grassmann manifold and generalized Jacobi polynomials — Part 2

Part 1 of this paper presented an explicit formula for generalized Jacobi polynomials of matrix argument. These polynomials constitute a complete system of orthogonal symmetric polynomials with respect to a multivariate beta measure. Zonal spherical functions on the Grassmann manifold may be expressed in terms of generalized Jacobi polynomials, and it is shown in Part 2 that they have an integral representation which generalizes the well-known integrals for Legendre and Gegenbauer polynomials of even order. In particular cases, this integral representation may be used to construct the zonal and associated spherical functions in terms of univariate special functions.

Spherical functions on the Grassmann manifold and generalized Jacobi polynomials — part 1

Generalized Jacobi polynomials constitute a complete system of orthogonal symmetric polynomials with respect to a multivariate beta measure. This paper gives an explicit formula for these polynomials in terms of the author's invariant polynomials in two matrix arguments, and derives the orthogonality property. Explicit formulas are also given for certain constants which appear in the theory of zonal polynomials, in cases where the partitional parameters have not more than two nonzero parts. Zonal spherical functions on the Grassmann manifold may be expressed in terms of generalized Jacobi polynomials. In Part 2 of this paper, it will be shown that these zonal spherical functions have an integral representation which generalizes the well-known integrals for Legendre and Gegenbauer polynomials of even order. In particular cases, this integral representation may be used to construct the zonal and associated spherical functions in terms of univariate special functions.