Discrete Semiclassical Orthogonal Polynomials of Class 2

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.

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.

Difference equations for discrete classical multiple 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.

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. >

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.

On a Difference Equation for Generalizations of Charlier Polynomials

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

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.

Power series expansion of a Hankel determinant

The Discrete Tchebichef Transform (DTT) is a transform method based on discrete orthogonal Tchebichef polynomials, which have applications found in image compression and video coding. Our method is to construct all DTT-related discrete orthogonal transforms in the required size (corresponding to the coding unit supported by H.266/VVC). To investigate the feature of Tchebichef polynomials, we make use of a novel discrete orthogonal matrix generation method with determined DTT roots, and scaling and rounding a DTT that depends on the quantization parameter, instead of integer approximation. We can obtain an accurate integer DTT matrix. Experimental results show that this method can improve the video quality and require fewer bit rates.

In an earlier work [Castillo et al., J. Math. Anal. Appl. 455, 1801–1821 (2017)], it was proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work, we use this fact to derive in a unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular, we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.