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33 pages, 447 KiB

Open AccessArticle

Multiple Meixner Polynomials on a Non-Uniform Lattice

by Jorge Arvesú and Andys M. Ramírez-Aberasturis

Mathematics 2020, 8(9), 1460; https://doi.org/10.3390/math8091460 - 31 Aug 2020

Cited by 2 | Viewed by 1545

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

34 pages, 376 KiB

Open AccessArticle

On Multivariate Bernstein Polynomials

by Mama Foupouagnigni and Merlin Mouafo Wouodjié

Mathematics 2020, 8(9), 1397; https://doi.org/10.3390/math8091397 - 21 Aug 2020

Cited by 11 | Viewed by 2997

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

9 pages, 267 KiB

Open AccessArticle

On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials

by Mohammad Masjed-Jamei, Nasser Saad, Wolfram Koepf and Fatemeh Soleyman

Mathematics 2020, 8(8), 1323; https://doi.org/10.3390/math8081323 - 8 Aug 2020

Cited by 1 | Viewed by 1543

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

15 pages, 337 KiB

Open AccessArticle

New Stability Criteria for Discrete Linear Systems Based on Orthogonal Polynomials

by Luis E. Garza, Noé Martínez and Gerardo Romero

Mathematics 2020, 8(8), 1322; https://doi.org/10.3390/math8081322 - 8 Aug 2020

Cited by 2 | Viewed by 2895

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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21 pages, 919 KiB

Open AccessArticle

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

by Carlos Hermoso, Edmundo J. Huertas, Alberto Lastra and Anier Soria-Lorente

Mathematics 2020, 8(8), 1300; https://doi.org/10.3390/math8081300 - 6 Aug 2020

Cited by 1 | Viewed by 2403

Abstract

This contribution deals with the sequence

{U_{n}^{(a)} (x; q, j)}_{n \geq 0}

of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal [...] Read more.

This contribution deals with the sequence

{U_{n}^{(a)} (x; q, j)}_{n \geq 0}

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 \in (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

U_{n}^{(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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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28 pages, 444 KiB

Open AccessArticle

On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

by Abey S. Kelil and Appanah R. Appadu

Mathematics 2020, 8(8), 1250; https://doi.org/10.3390/math8081250 - 31 Jul 2020

Cited by 4 | Viewed by 1815

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 |}^{α} exp (- c x^{6}), 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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

18 pages, 475 KiB

Open AccessArticle

On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle

by Jorge A. Borrego-Morell, Cleonice F. Bracciali and Alagacone Sri Ranga

Mathematics 2020, 8(7), 1161; https://doi.org/10.3390/math8071161 - 15 Jul 2020

Cited by 1 | Viewed by 1623

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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19 pages, 331 KiB

Open AccessArticle

On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle

by Lino G. Garza, Luis E. Garza and Edmundo J. Huertas

Mathematics 2020, 8(2), 246; https://doi.org/10.3390/math8020246 - 14 Feb 2020

Cited by 2 | Viewed by 2079

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

19 pages, 851 KiB

Open AccessArticle

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

by Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar and Richard Wellman

Mathematics 2020, 8(2), 182; https://doi.org/10.3390/math8020182 - 3 Feb 2020

Cited by 3 | Viewed by 1815

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

12 pages, 250 KiB

Open AccessFeature PaperArticle

A Characterization of Polynomial Density on Curves via Matrix Algebra

by Carmen Escribano, Raquel Gonzalo and Emilio Torrano

Mathematics 2019, 7(12), 1231; https://doi.org/10.3390/math7121231 - 12 Dec 2019

Cited by 1 | Viewed by 1715

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 [...] Read more.

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

16 pages, 356 KiB

Open AccessArticle

On Infinitely Many Rational Approximants to ζ(3)

by Jorge Arvesú and Anier Soria-Lorente

Mathematics 2019, 7(12), 1176; https://doi.org/10.3390/math7121176 - 3 Dec 2019

Cited by 1 | Viewed by 1593

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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20 pages, 514 KiB

Open AccessFeature PaperArticle

Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights

by Alejandro Arceo, Luis E. Garza and Gerardo Romero

Mathematics 2019, 7(9), 818; https://doi.org/10.3390/math7090818 - 5 Sep 2019

Cited by 6 | Viewed by 2474

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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33 pages, 444 KiB

Open AccessArticle

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

by Herbert Dueñas Ruiz, Francisco Marcellán and Alejandro Molano

Mathematics 2019, 7(2), 213; https://doi.org/10.3390/math7020213 - 25 Feb 2019

Cited by 1 | Viewed by 2283

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

Review

Jump to: Research

27 pages, 391 KiB

Open AccessReview

On Row Sequences of Hermite–Padé Approximation and Its Generalizations

by Nattapong Bosuwan

Mathematics 2020, 8(3), 366; https://doi.org/10.3390/math8030366 - 6 Mar 2020

Cited by 4 | Viewed by 1945

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

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

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