We observe that the zeros have regular configurations in the complex plane, which might be of independent interest. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given. Introduction to Real Orthogonal Polynomials. As motivation , consider the Dirichlet problem for the unit circle in the plane, which involves finding a harmonic function u r Orthogoy RMotion O0 bq :q x p.
Deyer, W. Gaussian quadrature for multiple orthogonal polynomials. We study multiple orthogonal polynomials of type I and type II, which have orthogonality conditions with respect to r measures. First we show a relation with the eigenvalue problem of a banded lower Hessenberg matrix Ln, containing the recurrence coefficients. As a consequence, we easily find that the multiple orthogonal polynomials of type I and type II satisfy a generalized Christoffel-Darboux identity. Furthermore, we explain the notion of multiple Gaussian quadrature for proper multi-indices , which is an extension of the theory of Gaussian quadrature for orthogonal polynomials and was introduced by Borges.
In particular, we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of Ln. Equivalences of the multi-indexed orthogonal polynomials. Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion.
Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only or type II indices only with shifted parameters.
Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. A connection between these two illustrations by means of complementary chain sequences is also observed.
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On multiple orthogonal polynomials for discrete Meixner measures. The paper examines two examples of multiple orthogonal polynomials generalizing orthogonal polynomials of a discrete variable, meaning thereby the Meixner polynomials. One example is bound up with a discrete Nikishin system, and the other leads to essentially new effects.
The limit distribution of the zeros of polynomials is obtained in terms of logarithmic equilibrium potentials and in terms of algebraic curves. Bibliography: 9 titles. Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, Stanton, Some basic hypergeometric polynomials arising from Some bas ic hypergeometr ic an a logues of the classical orthogonal polynomials and applications , to appear.
Golub , The This note considers the four classes of orthogonal polynomials --Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same….
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Orthogonal polynomials for refinable linear functionals. A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments.
Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. Numerical evidence suggests that it is also effective in floating-point arithmetic. Direct calculation of modal parameters from matrix orthogonal polynomials.
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The object of this paper is to introduce a new technique to derive the global modal parameter i. This contribution generalized the results given in Rolain et al.
Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-conditioned if not handled properly.
Communications of the Korean Mathematical Society
The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this paper a new approach is proposed to compute the system poles directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems. The proposed method will be compared with existing methods Van Der Auweraer and Leuridan  Chen and Xu . For this comparative study, simulated as well as experimental data will be used.
This paper presents a new adaptive control approach using Chebyshev orthogonal polynomials as basis functions in a least-squares functional approximation. The use of orthogonal basis functions improves the function approximation significantly and enables better convergence of parameter estimates. Flight control simulations demonstrate the effectiveness of the proposed adaptive control approach. Hermite polynomials and quasi- classical asymptotics.
Ali, S. Twareque, E-mail: twareque. We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi- classical asymptotics. Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture. Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials , 2D Legendre polynomials , Zernike square polynomials and Numerical polynomials.
They are all orthogonal over the full unit square domain. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness.
Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data. We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann-Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context local equilibrium.
For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the n particle dynamics. Asymptotic formulae for the zeros of orthogonal polynomials. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. Bibliography: 15 titles. Optimal approximation of harmonic growth clusters by orthogonal polynomials. Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoreticaI model for many physical processes, from the hydrodynamics of immiscible, viscous flows zero surface-tension limit of Hele-Shaw flows , to the granular dynamics of hard spheres, and even diffusion-limited aggregation.
Although a complete solution for the continuum case exists, efficient approximations of the boundary evolution are very useful due to their practical applications. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel is discussed, as well as relations to potential theory.
A note on the zeros of Freud-Sobolev orthogonal polynomials. We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials , i. Some numerical examples are shown. Crossover ensembles of random matrices and skew- orthogonal polynomials. Kumar, Santosh, E-mail: skumar. Kumar, A. Pandey, Phys. E, 79, , p. We start with Dyson's Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions.
For arbitrary dimensions we derive the joint probability density jpd of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles.
We focus on the orthogonal -unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew- orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter.
Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed. Perturbations of Jacobi polynomials and piecewise hypergeometric orthogonal systems. They are eigenfunctions of an exotic Sturm-Liouville boundary-value problem for the hypergeometric differential operator.
B 50 , and also used by our group Phys. B 55 , has numerical advantages of a pseudopotential technique while retaining the physics of an all-electron formalism. We describe a new method for generating the necessary set of atom-centered projector and basis functions, based on choosing the projector functions from a set of orthogonal polynomials multiplied by a localizing weight factor. We demonstrate the method by calculating the cohesive energies of CaF2 and Mo and the density of states of CaMoO4 which shows detailed agreement with LAPW results over a 66 eV range of energy including upper core, valence, and conduction band states.
Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived. Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality. We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials.