Sobolev Orthogonal Polynomials Research Articles

A study on Sobolev orthogonal polynomials on a triangle

A study on Sobolev orthogonal polynomials on a triangle

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

Sobolev orthogonal polynomials and spectral methods in boundary value problems

In the variational formulation of a boundary value problem for the harmonic oscillator, Sobolev inner products appear in a natural way. First, we study the sequences of Sobolev orthogonal polynomials with respect to such an inner product. Second, their representations in terms of a sequence of Gegenbauer polynomials are deduced as well as an algorithm to generate them in a recursive way is stated. The outer relative asymptotics between the Sobolev orthogonal polynomials and classical Legendre polynomials is obtained. Next we analyze the solution of the boundary value problem in terms of a Fourier-Sobolev projector. Finally, we provide numerical tests concerning the reliability and accuracy of the Sobolev spectral method.

Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ-Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role.

Coherent pairs of measures of the second kind on the real line and Sobolev orthogonal polynomials. An application to a Jacobi case

AbstractThe aim here is to consider the orthogonal polynomials with respect to an inner product of the type , where and is a coherent pair of positive measures of the second kind on the real line (CPPM2K on the real line). Properties of and the connection formulas they satisfy with the orthogonal polynomials associated with the measure ν0 are analyzed. It is also shown that the zeros of are the eigenvalues of a matrix, which is a single line modification of the Jacobi matrix associated with the measure ν0. The paper also looks at a special example of a CPPM2K on the real line, where one of the measures is the Jacobi measure, and provides a much more detailed study of the properties of the orthogonal polynomials and the corresponding connection coefficients. In particular, the relation that these connection coefficients have with the Wilson polynomials is exposed.

Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials

Given a pair of quasi-definite moment functionals {v0,v1} we introduce the concept of coherence of the second kind in terms of an algebraic relation that the corresponding sequences of orthogonal polynomials satisfy. We characterize such moment functionals and give some illustrative examples taking into account they are semiclassical of class at most one. The relation between the corresponding monic Jacobi matrices is stated. For a pair of moment functionals satisfying the coherence property of the second kind, a Sobolev inner product is introduced. The connection formulas between the sequence of monic orthogonal polynomials associated with such a Sobolev inner product and the sequence of monic polynomials orthogonal with respect to the moment functional v0 are given.

On the multiplication operator by an independent variable in matrix Sobolev spaces

We study the operator \(\mathcal {A}\) of multiplication by an independent variable in a matrix Sobolev space \(W^2(M)\). In the cases of finite measures on [a, b] with \((2\times 2)\) and \((3\times 3)\) real continuous matrix weights of full rank it is shown that the operator \(\mathcal {A}\) is symmetrizable. Namely, there exist two symmetric operators \(\mathcal {B}\) and \(\mathcal {C}\) in a larger space such that \(\mathcal {A} f = \mathcal {C} \mathcal {B}^{-1} f\), \(f\in D(\mathcal {A})\). As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These conditions involve two symmetric operators in an indefinite metric space.

On Sobolev orthogonal polynomials on a triangle

We use the invariance of the triangle T 2 = { ( x , y ) ∈ R 2 : 0 ⩽ x , y , 1 − x − y } \mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\} under the permutations of { x , y , 1 − x − y } \{x,y,1-x-y\} to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on T 2 \mathbf {T}^2 . These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.

Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation

Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation

The differential equation for Jacobi–Sobolev orthogonal polynomials with two linear perturbutions

We explicitly determine the higher-order differential equation for the Jacobi–Sobolev polynomials which extend the classical Jacobi polynomials with parameters α∈N0,β>−1 by two linear perturbations. More precisely, these polynomials are orthogonal with respect to a discrete Sobolev inner product which is associated with the Jacobi measure on the interval [−1,1] and two point masses at the right end point involving functions and their first derivatives. The corresponding Jacobi–Sobolev differential operator of order 4α+10 is appropriately defined by four elementary terms reflecting the influence of the point masses. Moreover, we prove that the new operator is symmetric with respect to the Sobolev inner product. This fundamental property readily implies the orthogonality of the Jacobi–Sobolev polynomials in the inner product space.

On multipliers for Fourier series in Sobolev orthogonal polynomials

Multipliers are studied for Fourier series in polynomials orthogonal in continuous-discrete Sobolev spaces. For the multiplier operator, existence results and norm estimates are obtained. The proofs are based on a representation of the Fejér kernel, the construction of a ‘humpbacked majorant’ and estimates for the norms of maximal functions. Bibliography: 45 titles.

Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Symmetric differential operators for Sobolev orthogonal polynomials of Laguerre- and Jacobi-type

ABSTRACT The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter and, in general, two point masses , S>0 at the origin involving functions and their first derivatives. Moreover we consider the Jacobi-Sobolev polynomials on the interval with Jacobi parameters and one Sobolev point mass S>0 at the right end-point of the domain. For , both polynomial systems are known to arise as eigenfunctions of certain spectral differential operators of finite order , provided that N = 0 in the Laguerre case. In this paper, new representations of the two differential operators are established which consist of a number of elementary components reflecting the influence of the parameters, appropriately. In particular, we show that the operators are symmetric with respect to the respective Sobolev inner products and so recover the orthogonality of the eigenfunctions.

A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: Bounds, asymptotics and zeros

This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with respect to these parameters. Criteria for knowing when the zeros of the (Sobolev) orthogonal polynomials and also the zeros of their derivatives stay within the unit disk have also been addressed. By numerical experiments some further information on the parameters is also found so that the zeros remain within the unit disk.

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ( $$SG$$ ), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. These orthogonal polynomials arise through the Gram–Schmidt orthogonalisation process applied on the set of monomials on $$SG$$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $$L^2$$ , $$L^\infty $$ , and Sobolev norms, and study their asymptotic behavior. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.

On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

For every system of OPRL or OPUC, we construct Sobolev orthogonal polynomials , with explicit integral representations involving . Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in n) and generalized eigenvalues of a differential operator (in z) are given. We define suitable Sobolev spaces with matrix weights and consider measurable factorizations of weights. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar spaces are discussed.

On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle

In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$. Differential equations of orders $\rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.

Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier–Sobolev projector allows to deal with very interesting approximation problems. The aim of this paper is twofold. First, we improve a well-known inequality by Lupaş by using connection formulas for Jacobi polynomials with different parameters. In the next step, we deduce Markov-type inequalities in weighted Sobolev spaces associated with generalized Laguerre and generalized Hermite weights.

On some classical type Sobolev orthogonal polynomials

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: 2F2(−n,1;α+1,κ+1;x) and 3F2(−n,n+α+β+1,1;α+1,κ+1;x) (α,β,κ>−1, n=0,1,…), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: Ly+λnDy=0, where L,D are linear differential operators with polynomial coefficients not depending on n. For nonnegative integer values of the parameter κ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained. Some generalizations of these polynomials are discussed as well.

On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

In the pioneering paper [13], the concept of Coherent Pair was introduced by Iserles et al. In particular, an algorithm to compute Fourier Coefficients in expansions of Sobolev orthogonal polynomials defined from coherent pairs of measures supported on an infinite subset of the real line is described. In this paper we extend such an algorithm in the framework of the so called Symmetric (1, 1)-Coherent Pairs presented in [8].