System Of Orthogonal Polynomials Research Articles

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann–Hilbert methods.

A proof of a conjecture about a symmetric system of orthogonal polynomials

ABSTRACTThe purpose of this note is to give an affirmative answer to a conjecture appearing in Berg [Open problems. Integral Transforms Spec Funct. 2015;26(2):90–95].

On the Modeling of Thin Bodies of Revolution

The new parametrization of the three-dimensional thin domain of an arbitrary body of rotation is considered, consisting in using several base surfaces in contrast to the classical approaches. The vector parametric equation is given. The geometric characteristics inherent to the new parameterization are determined. Expressions for the translation components of the unit tensor of the second rank and also relations connecting different bases and geometric characteristic generated by them are written. The determination of the moment of an arbitrary function is given. The motion equations and CR of the micropolar theory of elastic rotation bodies without a symmetry center of variable thickness are obtained under the new parametrization, from which equations are obtained for very thin and shallow rotation bodies, both variable and constant thickness. The equations of motion and CR of any approximation in the moments of unknown functions with respect to an arbitrary system of orthogonal polynomials and, in particular, for the system of Legendre polynomials are derived. The questions of setting initial-boundary problems are discussed.

To the Modeling of multilayer Thin Prismatic Bodies

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies were also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the fifth approximation of the micropolar theory of five-layer thin prismatic bodies are considered in more detail.

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on Z + \mathbb {Z}_+ to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.

On the differential equation for the Laguerre–Sobolev polynomials

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M,N>0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α∈N0, a spectral differential equation of finite order 4α+10. In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α,M,N. Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product.

On the Modeling of Five-Layer Thin Prismatic Bodies

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.

Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice

ABSTRACTIf is an orthogonal polynomial system, then satisfies a three-term recurrence relation of type with . On the other hand, Favard's theorem states that the converse is true. A general method to derive the coefficients , , in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or q-quadratic lattice is recalled. If a three-term recurrence relation is given as input, the Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmersau can identify its solution which is a (linear transformation of a) classical orthogonal polynomial system of a continuous, a discrete or a q-discrete variable, if applicable. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or q-quadratic lattice. Motivated by an open problem, submitted by Alhaidari during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications, which will serve as application, the Maple implementation retode of Koepf and Schmersau is extended to cover classical orthogonal polynomial solutions on quadratic or q-quadratic lattices of three-term recurrence relations.

Sparse learning of partial differential equations with structured dictionary matrix.

This paper presents a "structured" learning approach for the identification of continuous partial differential equation (PDE) models with both constant and spatial-varying coefficients. The identification problem of parametric PDEs can be formulated as an ℓ1/ℓ2-mixed optimization problem by explicitly using block structures. Block-sparsity is used to ensure parsimonious representations of parametric spatiotemporal dynamics. An iterative reweighted ℓ1/ℓ2 algorithm is proposed to solve the ℓ1/ℓ2-mixed optimization problem. In particular, the estimated values of varying coefficients are further used as data to identify functional forms of the coefficients. In addition, a new type of structured random dictionary matrix is constructed for the identification of constant-coefficient PDEs by introducing randomness into a bounded system of Legendre orthogonal polynomials. By exploring the restricted isometry properties of the structured random dictionary matrices, we derive a recovery condition that relates the number of samples to the sparsity and the probability of failure in the Lasso scheme. Numerical examples, such as the Schrödinger equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, the Burger equation, and the Fisher equation, suggest that the proposed algorithm is fairly effective, especially when using a limited amount of measurements.

Modeling of multilayer thin bodies

In this work, we considered the new parametrization of a multilayer thin domain. In particular, in contrast to classic approaches, we used several base surfaces and an analytic method with the application of orthogonal polynomial systems. We gave the vector parametric equation of each layer and the system of vector parametric equations of a multilayer thin domain and introduced the geometric characteristics for the proposed parametrization. We also derived the expressions for the transfer components of the second-rank identity tensor and the relations connecting the various families of bases and presented some differential operators, the system of equations of motion, the heat flow equation, the constitutive relations of the theory of the micropolar elasticity, and the Fourier heat conduction law under this parametrization of the thin-body domain. Finally, we gave the classification and statements of boundary value problems in the theory of thin bodies.

Ultraspherical moments on a set of disjoint intervals

Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.

A Bochner type characterization theorem for exceptional orthogonal polynomials

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles.

The higher-order differential operator for the generalized Jacobi polynomials — new representation and symmetry

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder’s generalized Jacobi polynomials with four parameters α,β∈N0 and M,N≥0 determining the orthogonality measure on the interval −1≤x≤1. The corresponding differential equation of order 2α+2β+6 is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.

D-Orthogonal Analogs of Classical Orthogonal Polynomials

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.

Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

Let [Formula: see text] denote a non-commutative monotone Lévy process. Let [Formula: see text] denote the corresponding monotone Lévy noise, i.e. formally [Formula: see text]. A continuous polynomial of [Formula: see text] is an element of the corresponding non-commutative [Formula: see text]-space [Formula: see text] that has the form [Formula: see text], where [Formula: see text]. We denote by [Formula: see text] the space of all continuous polynomials of [Formula: see text]. For [Formula: see text], the orthogonal polynomial [Formula: see text] is defined as the orthogonal projection of the monomial [Formula: see text] onto the subspace of [Formula: see text] that is orthogonal to all continuous polynomials of [Formula: see text] of order [Formula: see text]. We denote by [Formula: see text] the linear span of the orthogonal polynomials. Each orthogonal polynomial [Formula: see text] depends only on the restriction of the function [Formula: see text] to the set [Formula: see text]. The orthogonal polynomials allow us to construct a unitary operator [Formula: see text], where [Formula: see text] is an extended monotone Fock space. Thus, we may think of the monotone noise [Formula: see text] as a distribution of linear operators acting in [Formula: see text]. We say that the orthogonal polynomials belong to the Meixner class if [Formula: see text]. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: [Formula: see text] and [Formula: see text]. In this case, the monotone Lévy noise has the representation [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are the (formal) creation and annihilation operators at [Formula: see text] acting in [Formula: see text].

Vector Orthogonal Polynomials with Bochner’s Property

Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre, and Bessel have the property that the polynomials of each system are eigenfunctions of a second-order ordinary differential operator. According to a classical theorem by Bochner, they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk, and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard–Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem, and this characterizes them. Motivated by Bochner’s theorem, we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner’s property. The goal of this paper is to introduce methods for construction of VOP which have Bochner’s property. The methods are purely algebraic and are based on automorphisms of noncommutative algebras. They also use ideas from the so-called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations.

Three systems of orthogonal polynomials and L2-boundedness of two associated operators

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L2-spaces. However, in subsequent papers, we intend to extend the study to Lp-spaces (1<p<∞).

On linear spectral transformations and the Laguerre–Hahn class

ABSTRACTWe study the Christoffel and Geronimus transformations for Laguerre–Hahn orthogonal polynomials on the real line. It is analysed the modification on the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recurrence relation coefficients.

An analogue of the big q-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.

Jacobi Operators and Orthonormal Matrix-Valued Polynomials. I

It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator. A system of orthogonal operator-valued polynomials is constructed.