Discrete orthogonal polynomials and difference equations of several variables

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

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.

Difference equations for discrete classical multiple orthogonal polynomials

In this paper we shall derive energy integral inequalities for the linear hyperbolic partial differential equation of any order in several variables with multiple characteristics and constant coefficients. In [1] Garding uses an idea by Leray [6] to develop energy integrals for totally hyperbolic equations (including variable coefficients). Totally hyperbolic equations are subject to the severe restriction that all characteristics remain distinct(2). In his method G'arding multiplies the given operator of order n by another operator of order n 1 whose characteristics separate those of the given operator. Using the Fourier transform and Parseval's formula he arrives at the energy inequality, which is an a priori estimate of the square integral of the solution of the given equation and all the derivatives up to order n -1. In our case of an equation with multiple characteristics we use a variant of G'arding's method. Our energy inequality is of a different form in as much as it provides an a priori estimate of the integral of the square of the solution and certain non-negative integral forms of its derivatives up to order n -1. However, we also show that for totally hyperbolic equations our energy inequalities reduce to the standard form. In hyperbolic equations with multiple characteristics the influence of the lower order terms must be taken into account and hence the form of these lower order terms is suitably restricted. In totally hyperbolic equations no restrictions are imposed on the lower order terms. Our definition of hyperbolicity is therefore such that in the case of distinct characteristics no restrictions are imposed on the lower order terms. In [2] Gairding has given a criterion for hyperbolicity of equations in several variables with multiple characteristics and constant coefficients. A. Lax [4] extended this criterion, in the case of two independent variables, to include variable coefficients. We shall not be concerned here with

On a Difference Equation for Generalizations of Charlier Polynomials

We study discrete semiclassical orthogonal polynomials of class s = 1. By considering particular solutions of the Pearson equation, we obtain five canonical families of such polynomials. We also consider limit relations between these and other families of orthogonal polynomials.

Modeling and Simulation Technology has become an important means to study various complex systems with its extensive application. Thus, the accuracy of the simulation models becomes a critical problem and needs to be assessed by employing an appropriate model validation method. The simulation models often have multivariate dynamic responses with uncertainty, while most of the existing validation methods concentrate on the validation of the static responses. Hence, a new validation method is proposed in this paper to validate the dynamic responses of the simulation models over the time domain at a single validation site and multiple validation sites through introducing the discrete Chebyshev polynomials and area metric. For each time series, the orthogonal expansion coefficients are extracted primarily by representing the time series with the discrete orthogonal polynomials. Then, the area metric and the [Formula: see text]-pooling metric are employed to validate all the uncorrelated coefficients at a single validation site and multiple validation sites, respectively, and the final validation result is obtained by summarizing the metric values. The feasibility and effectiveness of the proposed model validation method are illustrated through the example of the terminal guidance stage of the flight vehicle.

Attention is drawn to a phenomenon of “pseudostability” in connection with the three-term recurrence relation for discrete orthogonal polynomials. The computational implications of this phenomenon are illustrated in the case of discrete Legendre and Krawtchouk polynomials. The phenomenon also helps to explain a form of instability in Stieltjes's procedure for generating recursion coefficients of discrete orthogonal polynomials.

Spectral synthesis and spectral analysis.- Spectral analysis and spectral synthesis on discrete Abelian groups.- Spectral synthesis and functional equations.- Mean periodic functions.- Difference equations in several variables.- Spectral analysis and synthesis on polynomial hypergroups in a single variable.- Spectral analysis and synthesis on multivariate polynomial hypergroups.

Zero distributions for discrete orthogonal polynomials

This investigation evaluated the application of discrete orthogonal polynomial deconvolution (DOPD) to the restoration of two dimensional images degraded by separable spatially-varying point spread functions. The one-dimensional DOPD techniques developed for time-varying systems easily extend to the separable, spatially-varying two-dimensional case. The quality of the restoration depends on the basis set selection and on the characteristics of noise in the degraded image. This method of restoration is noniterative, and does not require matrix inversion or segmentation of the image into locally invariant PSF regions. The primary shortcoming is that the quality of the restored image depends on the adequacy of representation of the original image by a limited basis set. Initial experimental results with severely degraded noisy images are promising. Further investigation will establish the scope of practical image restoration applications. >

Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters

Difference equations in several variables

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals.While existing characterisations could only cover time complexity or were restricted to functions over the integers, here we deal with real numbers and also cover space complexity. Furthermore, we prove that no artificial sign or test function is needed even for time complexity.At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner productwhere ω is a weight function, ξ ≤ 0, and M, N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function ω.In particular, for ω(x) = x α e -x we obtain the asymptotics for discrete Laguerre-Sobolev orthogonal polynomials.