System Of Orthogonal Polynomials Research Articles

Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices

We study the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian Unitary Ensemble (GUE) of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near $\lambda_{\max}$, $\rho_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $\lambda_{\max}$ and (ii) the probability density function of the gap between the first two largest eigenvalues, $p_{\rm GAP}(r,N)$. In the edge scaling limit where $r = {\cal O}(N^{-1/6})$, which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that $\rho_{\rm DOS}(r,N)$ and $p_{\rm GAP}(r,N)$ are characterized by scaling functions which can be expressed in terms of the solution of a Lax pair associated to the Painlev\'e XXXIV equation. This provides an alternative and simpler expression for the gap distribution, which was recently studied by Witte, Bornemann and Forrester in Nonlinearity 26, 1799 (2013). Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.

Asymptotics of linear recurrences

In this paper, we review the asymptotic results that are now available for second-order linear difference equations containing a parameter (or, equivalently, three-term recurrence relations). These include asymptotic expansions for solutions to these equations when the parameter is fixed or varying in an interval containing a turning point or a transition point. Also presented is a method for deriving asymptotic approximations for solutions when the initial values are given. These results are particularly useful when a given system of orthogonal polynomials (i) does not satisfy any second-order differential equation, (ii) does not have any integral representation, and (iii) is not even associated with a unique (or any) weight function.

Characterizations of distributional weights for weak orthogonal polynomials satisfying a second-order differential equation

Let \(\{ P_n(x) \}_{n=0}^\infty \) be a monic weak orthogonal polynomial system (weak OPS) of order \(m\) relative to \( \sigma \). Then we first prove that \( \sigma \) satisfies a new moment equation $$\begin{aligned} P_{m+1} (x) \sigma =0, \end{aligned}$$ (0.1) and then we find a general representation of \(\sigma \) from the moment equation (0.1). Assuming moreover that \(\{ P_n(x) \}_{n=0}^\infty \) satisfies a second-order differential equation, we find a distributional representation of \( \sigma \) for such weak OPSs by the coefficients \(\ell _i\)’s and the eigenvalues \(\lambda _k\) of differential equation. By analyzing those moment equations thoroughly, we finally classify all moment functionals \( \sigma \) for the weak OPSs satisfying a second-order differential equation.

Prolate spheroidal wavelets and multidimensional chromatic series expansions

Chromatic series were originally introduced for bandlimited functions. The nth chromatic derivative of an analytic function is a linear combination of kth ordinary derivatives with 0 ≤ k ≤ n, where the coefficients of linear combination are based on a suitable system of orthogonal polynomials. Chromatic derivative and series expansions of bandlimited functions have been used as a replacement for Taylor's series and they have been shown to be more useful in practical signal processing applications than Taylor series. In this paper we have shown that the theory can be extended to prolate spheroidal wavelet series that than combine chromatic series with sampling series in higher dimensions. The multidimensional case has much reacher structure than in the univariate case and will find more applications in image processing and analysis.

Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux–Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2 m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2ℓ + 3 recurrence relation where ℓ is the length of the partition λ. Explicit expressions for such recurrence relations are given.

Kronrod extensions with multiple nodes of quadrature formulas for Fourier coefficients

We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives, is considered. We prove the existence and uniqueness of Kronrod extensions with multiple nodes of standard Gaussian quadrature formulas with multiple nodes for several weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coefficients. For the quadrature formulas for the Fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct Kronrod extensions with multiple nodes and highest algebraic degree of precision. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.

On solvability and integrability of the Rabi model

The quasi-exactly solvable Rabi model is investigated within the framework of the Bargmann Hilbert space of analytic functions B. On applying the theory of orthogonal polynomials, the eigenvalue equation and eigenfunctions are shown to be determined in terms of three systems of monic orthogonal polynomials. The formal Schweber quantization criterion for an energy variable x, originally expressed in terms of infinite continued fractions, can be recast in terms of a meromorphic function F(z)=a0+∑k=1∞Mk/(z−ξk) in the complex plane C with real simple poles ξk and positive residues Mk. The zeros of F(x) on the real axis determine the spectrum of the Rabi model. One obtains at once that, on the real axis, (i) F(x) monotonically decreases from +∞ to −∞ between any two of its subsequent poles ξk and ξk+1, (ii) there is exactly one zero of F(x) for x∈(ξk,ξk+1), and (iii) the spectrum corresponding to the zeros of F(x) does not have any accumulation point. Additionally, one can provide a much simpler proof that the spectrum in each parity eigenspace B± is necessarily nondegenerate. Thereby the calculation of spectra is greatly facilitated. Our results allow us to critically examine recent claims regarding solvability and integrability of the Rabi model.

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.

The Eneström–Kakeya theorem encounters the theory of orthogonal polynomials on the unit circle

A classical result due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. The main subject of the present paper is a study of this fact with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we will be interested in the special case, where the zeros of the polynomials in question are not located on the boundary of the estimate which occurs in the Eneström–Kakeya theorem. Among other things, we will give characterizations of this case in terms of orthogonal polynomials and present an alternative approach to a well-known characterization due to Hurwitz. Furthermore, we will give some insight how one can apply the main results of this paper in the context of positive Hermitian Toeplitz matrices.

Sharp estimates of the Jacobi heat kernel

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters above -1/2. As an application of the upper bound obtained, we show that the maximal operator of the multi-dimensional Jacobi heat semigroup satisfies a weak type (1; 1) inequality. We also obtain sharp estimates of the Poisson-Jacobi kernel.

Structure Variation Hyper Arithmetical Sort Operators and Applications

Most of the better-known precisely resolvable Schrodinger equations are directly associated with these operators and every arithmetical kind equation defines either a finite or Associate in Nursing infinite system of orthogonal polynomials. Its attainable that everyone these operators are often studied along in an exceedingly unified mathematical formalism The corresponding associated special functions are Eigen functions of some form invariant operators, and most of the formula occurring during this study of precisely resolvable quantum systems follow from tiny low range of elementary mathematical results. The systematic study recovers better-known ends up in a natural unified manner, and permits one to increase ways and results better-known specially cases.

Construction of a Lax Pair for the E6(1)q-Painlevé System

We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6 $ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.

A Conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

Weak lensing tomography with orthogonal polynomials

The topic of this article is weak cosmic shear tomography where the line of sight-weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed TaRDiS (Tomography with orthogonAl Radial Distance polynomIal Systems). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher-matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomography are made easier to understand.

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

Multiplicity function for tensor powers of modules of the A n algebra

We consider the decomposition of the pth tensor power of the module \(L^{\omega _1 }\) over the algebra An into irreducible modules, \((L^{\omega _1 } )^{ \otimes p} = \sum\nolimits_v {m(v,p)L^v }\). This problem occurs, for example, in finding the spectrum of an invariant Hamiltonian of a spin chain with p nodes. To solve the problem, we propose using the Weyl symmetry properties. For constructing the coefficients m(ν, p) as functions of p, we develop an algorithm applicable to powers of an arbitrary module. We explicitly write an expression for the multiplicities m(ν, p) in the decomposition of powers of the first fundamental module of sl(n+1). Based on the obtained results, we find new properties of systems of orthogonal polynomials (multivariate Chebyshev polynomials). Our algorithm can also be applied to tensor powers of modules of other simple Lie algebras.

Orthogonal polynomials in the normal matrix model with a cubic potential

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder.The main goal of this paper is to develop the Riemann–Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures (μ1,μ2) on the complex plane. We then formulate a 3×3 matrix valued RH problem for the orthogonal polynomials in hand, and we apply the nonlinear steepest descent method of Deift–Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure (μ1,μ2), and the construction of a global parametrix.The main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure μ1, the first component of the equilibrium measure. We also obtain analytical results for the measure μ1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve.

Sparse Legendre expansions via [formula omitted]-minimization

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m≍slog4(N) random samples that are chosen independently according to the Chebyshev probability measure dν(x)=π−1(1−x2)−1/2dx. As an efficient recovery method, ℓ1-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.

Fredholm Determinant Evaluations of the Ising Model Diagonal Correlations and their λ Generalization

The diagonal spin–spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants—one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ‐parameter generalization and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case [1], applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi‐orthogonal situation.

N-symmetric Chebyshev polynomials in a composite model of a generalized oscillator

We continue to study a composite model of a generalized oscillator generated by an N-periodic Jacobi matrix. The foundation of the model is a system of orthogonal polynomials connected to this matrix for N = 3, 4, 5. We show that such polynomials do not exist for N ≥ 6.