Special Issue "Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 December 2020).

Special Issue Editors

Prof. Dr. Francisco Marcellan
E-Mail Website
Guest Editor
Universidad Carlos III de Madrid, Departamento de Matemáticas, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain
Interests: orthogonal polynomials; moment problems; distribution of zeros; integrable systems; random matrices; stochastic processes; signal theory; quadrature formulas; spectral methods for boundary value problems; fourier expansions; structured matrices; integral transforms please revise intereest of Edmundo Huertas orthogonal polynomials; moment problems; distribution of zeros; integrable systems; random matrices; stochastic processes; signal theory; quadrature formulas; spectral methods for boundary value problems; fourier expansions; structured matrices; integral transforms
Dr. Edmundo Huertas
E-Mail Website
Guest Editor
Dpto. de Física y Matemáticas, Universidad de Alcalá (UAH), Ctra. Madrid-Barcelona, Km. 33,600, Alcalá de Henares, 28805 Madrid, Spain
Interests: orthogonal polynomials; moment problems; distribution of zeros; integrable systems; random matrices; stochastic processes; signal theory; quadrature formulas; spectral methods for boundary value problems; fourier expansions; structured matrices; integral transforms

Special Issue Information

Dear Colleagues,

In recent years, the theory of orthogonal polynomials has received a great amount of interest because of its wide role in Pure and Applied Mathematics. Orthogonal polynomials are essential tools for the solution of many problems in the spectral theory of differential and difference equations, Painlevé equations (discrete and continuous versions), numerical methods in quadrature on the real line and the unit circle, as well as cubature formulas on multidimensional domains, with applications ranging from Number Theory to Approximation Theory, Combinatorics to Group representation, integrable systems, random matrices, and linear system theory to signal processing.

The aims of the proposed Special Issue are:

  • To show some recent trends in the research on orthogonal polynomials, with a special emphasis on their analytic properties and approximation theory. Different examples of orthogonality (Sobolev, multiple, multivariate, matrix) will be studied, as well as the asymptotic properties of the corresponding sequences of orthogonal polynomials and the behavior of their zeros;
  • To emphasize their impact in Mathematical Physics, mainly in integrable systems and Painlevé equations (discrete and continuous cases), as they are strongly related to the coefficients of three term relation, satisfied by a sequence of orthogonal polynomials and time-depending measures supported on the real line.

Prof. Dr. Francisco Marcellan
Dr. Edmundo Huertas
Guest Editors

Manuscript Submission Information

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Keywords

  • Orthogonal polynomials on the real line
  • Orthogonal polynomials on the unit circle
  • Matrix orthogonal polynomials
  • Multiple orthogonal polynomials
  • Multivariate orthogonal polynomials
  • Sobolev orthogonal polynomials
  • Integrable systems
  • Random matrices
  • Quadrature and cubature formulas
  • Rational approximation
  • Approximation with splines
  • Wavelets

Published Papers (14 papers)

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Research

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Open AccessArticle
Multiple Meixner Polynomials on a Non-Uniform Lattice
Mathematics 2020, 8(9), 1460; https://doi.org/10.3390/math8091460 - 31 Aug 2020
Viewed by 469
Abstract
We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These [...] Read more.
We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained. Full article
Open AccessArticle
On Multivariate Bernstein Polynomials
Mathematics 2020, 8(9), 1397; https://doi.org/10.3390/math8091397 - 21 Aug 2020
Viewed by 598
Abstract
In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, [...] Read more.
In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions. Full article
Open AccessArticle
On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials
Mathematics 2020, 8(8), 1323; https://doi.org/10.3390/math8081323 - 08 Aug 2020
Cited by 1 | Viewed by 520
Abstract
Using the Sturm–Liouville theory in q-difference spaces, we prove the finite orthogonality of q-Pseudo Jacobi polynomials. Their norm square values are then explicitly computed by means of the Favard theorem. Full article
Open AccessArticle
New Stability Criteria for Discrete Linear Systems Based on Orthogonal Polynomials
Mathematics 2020, 8(8), 1322; https://doi.org/10.3390/math8081322 - 08 Aug 2020
Viewed by 536
Abstract
A new criterion for Schur stability is derived by using basic results of the theory of orthogonal polynomials. In particular, we use the relation between orthogonal polynomials on the real line and on the unit circle known as the Szegő transformation. Some examples [...] Read more.
A new criterion for Schur stability is derived by using basic results of the theory of orthogonal polynomials. In particular, we use the relation between orthogonal polynomials on the real line and on the unit circle known as the Szegő transformation. Some examples are presented. Full article
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Open AccessArticle
On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order
Mathematics 2020, 8(8), 1300; https://doi.org/10.3390/math8081300 - 06 Aug 2020
Viewed by 672
Abstract
This contribution deals with the sequence {Un(a)(x;q,j)}n0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality. Full article
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Open AccessArticle
On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight
Mathematics 2020, 8(8), 1250; https://doi.org/10.3390/math8081250 - 31 Jul 2020
Cited by 1 | Viewed by 583
Abstract
Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical [...] Read more.
Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(cx6),c>0,α>1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials. Full article
Open AccessArticle
On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle
Mathematics 2020, 8(7), 1161; https://doi.org/10.3390/math8071161 - 15 Jul 2020
Cited by 1 | Viewed by 485
Abstract
We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit [...] Read more.
We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen–Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen–Morse potential, an identity involving Gegenbauer polynomials is obtained. Full article
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Open AccessArticle
On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle
Mathematics 2020, 8(2), 246; https://doi.org/10.3390/math8020246 - 14 Feb 2020
Cited by 2 | Viewed by 599
Abstract
In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach [...] Read more.
In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle. Full article
Open AccessArticle
Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials
Mathematics 2020, 8(2), 182; https://doi.org/10.3390/math8020182 - 03 Feb 2020
Viewed by 566
Abstract
In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in [...] Read more.
In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in the corresponding eigenvalues, more exactly, in their asymptotic behavior. Thus, we can determine a limit value which links this asymptotic behavior and the uniform norm of the orthonormal polynomials in a logarithmic scale. This value appears in the theory of reproducing kernel Hilbert spaces. On the other hand, we tackle a more general case than the one considered in the literature previously. Full article
Open AccessFeature PaperArticle
A Characterization of Polynomial Density on Curves via Matrix Algebra
Mathematics 2019, 7(12), 1231; https://doi.org/10.3390/math7121231 - 12 Dec 2019
Viewed by 572
Abstract
In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices. Full article
Open AccessArticle
On Infinitely Many Rational Approximants to ζ(3)
Mathematics 2019, 7(12), 1176; https://doi.org/10.3390/math7121176 - 03 Dec 2019
Cited by 1 | Viewed by 528
Abstract
A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and [...] Read more.
A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated. Full article
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Open AccessFeature PaperArticle
Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights
Mathematics 2019, 7(9), 818; https://doi.org/10.3390/math7090818 - 05 Sep 2019
Cited by 2 | Viewed by 744
Abstract
In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect [...] Read more.
In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented. Full article
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Open AccessArticle
A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals
Mathematics 2019, 7(2), 213; https://doi.org/10.3390/math7020213 - 25 Feb 2019
Cited by 1 | Viewed by 944
Abstract
In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described. Full article

Review

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Open AccessReview
On Row Sequences of Hermite–Padé Approximation and Its Generalizations
Mathematics 2020, 8(3), 366; https://doi.org/10.3390/math8030366 - 06 Mar 2020
Cited by 1 | Viewed by 601
Abstract
Hermite–Padé approximation has been a mainstay of approximation theory since the concept was introduced by Charles Hermite in his proof of the transcendence of e in 1873. This subject occupies a large place in the literature and it has applications in different subjects. [...] Read more.
Hermite–Padé approximation has been a mainstay of approximation theory since the concept was introduced by Charles Hermite in his proof of the transcendence of e in 1873. This subject occupies a large place in the literature and it has applications in different subjects. Most of the studies of Hermite–Padé approximation have mainly concentrated on diagonal sequences. Recently, there were some significant contributions in the direction of row sequences of Type II Hermite–Padé approximation. Moreover, various generalizations of Type II Hermite–Padé approximation were introduced and studied on row sequences. The purpose of this paper is to reflect the current state of the study of Type II Hermite–Padé approximation and its generalizations on row sequences. In particular, we focus on the relationship between the convergence of zeros of the common denominators of such approximants and singularities of the vector of approximated functions. Some conjectures concerning these studies are posed. Full article
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