Sequence Of Monic Orthogonal Polynomials Research Articles

Properties and Convergence Analysis of Orthogonal Polynomials, Reproducing Kernels, and Bases in Hilbert Spaces Associated with Norm-Attainable Operators

This research paper delves into the properties and convergence behaviors of various sequences of orthogonal polynomials, reproducing kernels, and bases within Hilbert spaces governed by norm-attainable operators. Through rigorous analysis, the study establishes the completeness of the sequences of monic orthogonal polynomials and orthonormal polynomials, highlighting their comprehensive representation and approximation capabilities in the Hilbert space. The paper also demonstrates the completeness and density attributes of the sequence of normalized reproducing kernels, showcasing its effective role in capturing the intrinsic structure of the space. Additionally, the research investigates the uniform convergence of these sequences, revealing their convergence to essential operators within the Hilbert space. Ultimately, these results contribute to both theoretical understanding and practical applications in various fields by providing insights into function approximation and representation within this mathematical framework.

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.

Characterization of the D-Laguerre–Hahn orthogonal polynomials of class one via the cubic decomposition

In the present work, we deal with a monic orthogonal polynomial sequence (MOPS) deduced by its cubic decomposition (CD) . Our goal is to determine all D-Laguerre–Hahn strict of class one of these sequences. From the CD structure, only one family of MOPS appears in connection with the D-Laguerre–Hahn of class zero of Jacobi type in the singular case and the nonsingular one. Their recurrence coefficients are highlighted.

Other family of cubic decomposition of a semiclassical form of class one

In the present work, we are interested in the study of a family of monic orthogonal polynomial sequence deduced by its cubic decomposition (CD) , . Our objective is to give some algebraic properties on its connecting sequences derived by the CD and essentially to deal with semiclassical character of this family of sequences. From the CD structure, only five semiclassical MOPS of class one appear, which their recurrence coefficients are explicitly given. An integral representation of positive definite case is established.

Associated orthogonal polynomials of the first kind and Darboux transformations

Let u be a quasi-definite linear functional defined on the linear space of polynomials P with complex coefficients. For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) (Pn)n∈N, which satisfies a three term recurrence relation. If we increase the indices of the recurrence relation by one unity, then we get the sequence of associated polynomials of the first kind as solution. These polynomials will be orthogonal with respect to a linear functional denoted by u(1). In the literature two special transformations of the functional u are studied, the canonical Christoffel transformation u˜=(x−c)u and the canonical Geronimus transformation uˆ=(x−c)−1u+Mδc, where c is a fixed complex number, M is a free complex parameter and δc is the linear functional defined on P as 〈δc,p(x)〉=p(c). For the Christoffel transformation with SMOP (P˜n)n∈N, we are interested in analyzing the relation between the linear functionals u(1) and u˜(1). The super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind (Pn(1))n∈N and (P˜n(1))n∈N, respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between u−1 (the inverse functional) and u(1) which can be given from a quadratic Geronimus transformation.

Spectral Transformations and Associated Linear Functionals of the First Kind

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences

In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials.

Some New Connection Relations Related to Classical Orthogonal Polynomials

In this paper, we deal with a problem of positivity of linear functionals in the linear space ℙ of polynomials in one variable with complex coefficients. Some new results of connection relations between the corresponding sequences of monic orthogonal polynomials of classical character are established.

Third-degree semiclassical forms of class one arising from cubic decomposition

ABSTRACT An orthogonal polynomial sequence with respect to a regular form (linear functional) w is said to be semiclassical if there exists a monic polynomial φ and a polynomial ψ with , such that . Recently, all semiclassical monic orthogonal polynomial sequences , of class one obtained from the cubic decompositions (CD) satisfying the relation have been determined [see Tounsi MI, Bouguerra I. Cubic decomposition of a family of semiclassical polynomial sequences of class one. Integral Transforms Spec Funct. 2015;26(5):377–394; Castillo K, de Jesus MN, Petronilho J. On semiclassical orthogonal polynomials via polynomial mappings. J Math Anal Appl. 2017;455(2):1801–1821]. The aim of our work is to study semiclassical sequences of the above family such that their corresponding Stieltjes function satisfies a cubic relation of the form , where A, B, C, D are polynomials. In particular, the link between w and the Jacobi form is established. Furthermore, both the characteristic elements of the structure relation and of the second-order differential equation are explicitly given.

2 × 2 hypergeometric operators with diagonal eigenvalues

In this work we give all the order-two hypergeometric operators D, symmetric with respect to some 2 × 2 irreducible matrix-weight W on (0,1) such that DPn=Pnλn00μn with no repetition among the eigenvalues {λn,μn}n∈N0, where {Pn}n∈N0 is the (unique) sequence of monic orthogonal polynomials with respect to W.We obtain a new family of such operators and weights, depending on three parameters, generalizing some older examples. We also give, in a very explicit way, the corresponding monic orthogonal polynomials, their three term recurrence relation and their squared matrix-norms.

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs

In this paper we will discuss asymptotic properties of monic polynomials {Snλ(x)}n≥0 orthogonal with respect to the Sobolev inner product〈p,q〉S=∫Rp(x)q(x)dμ0+λ∫Rp′(x)q′(x)dμ1, with λ>0, dμ0=e−x2dx, dμ1=x2+ax2+be−x2dx, a,b∈R+ and a≠b. It is well known that (μ0,μ1) is a pair of symmetric (1,1)-coherent measures. This means that there exist sequences {an}n∈N, {bn}n∈N, an≠bn for every n∈N, such that the algebraic relationHn(x)+bn−2Hn−2(x)=Qn(x)+an−2Qn−2(x),n≥2, is satisfied, where {Qn(x)}n≥0 is the sequence of monic orthogonal polynomials associated with μ1 and {Hn(x)}n≥0 is the sequence of monic Hermite polynomials. We will study the relative asymptotics for Sobolev scaled polynomials and we will obtain Mehler–Heine type formulas, among others.

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c j } =0 be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P j } =0 be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P j } =0 . We also obtain an expression for the corresponding second kind polynomials.

Cubic decomposition of a family of semiclassical orthogonal polynomials of class two

ABSTRACTWe deal with a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition , . Only four monic orthogonal polynomial sequences (MOPS) appear in which their recurrence coefficients are explicitly given.

Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences $$\{T_n\}_{n\ge 0}$$ and $$\{U_n\}_{n\ge 0}$$ , the Chebyshev polynomials of the first and second kind, satisfy the relation $$DT_{n+1}=(n+1)U_n$$ ( $$n\ge 0$$ ). One can also easily check that the following “inverse” of the mentioned formula holds: $${\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}$$ ( $$n\ge 0$$ ), where $${\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D$$ with $$\xi $$ being an arbitrary nonzero parameter and $${\mathbb {I}}$$ representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves $${\mathcal {U}}_\xi $$ which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence $$\{a^{-n}U_n(ax)\}_{n\ge 0}$$ where $$a^2=-\xi ^{-1}$$ , is actually the only monic orthogonal polynomial sequence which is $${\mathcal {U}}_\xi $$ -classical (i.e. for which the application of the raising operator $${\mathcal {U}}_\xi $$ turns the original sequence into another orthogonal one).

Orthogonal polynomial approximation method for stability prediction in milling

Based on orthogonal polynomial approximation scheme, this paper presents several stability prediction methods using different kinds of orthogonal polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). Firstly, this work employs the classical Legendre and Chebyshev polynomials to approximate the state term, delayed term, and periodic-coefficient matrix. With the help of direct integration scheme (DIS), the state transition matrixes which indicate the mapping relations of the dynamic response between the current tooth pass and the previous tooth pass are obtained. The stability lobe diagrams for single degree of freedom (DOF) and two DOF milling models are generated by using the Legendre and Chebyshev polynomial approximation-based methods. The rate of convergence of the Legendre and Chebyshev polynomial-based methods is compared with that of the benchmark first-order semi-discretization method (1stSDM). The comparison results indicate that the rate of convergence and the numerical stability of the Legendre and Chebyshev polynomial-based methods are both need to be improved. In order to develop new methods with high rate of convergence and numerical stability base on DIS, the monic orthogonal polynomial sequences are constructed by using Gram-Schmidt orthogonalization to approximate the state term, delayed term, and periodic-coefficient matrix. The rate of convergence and the computational efficiency of the monic orthogonal polynomial-based methods are evaluated by comparing with those of the benchmark 1stSDM. The results turn out that the monic orthogonal polynomial-based methods are advantageous in terms of the rate of convergence and numerical stability. The stability lobe diagrams for single DOF and two DOF milling models obtained by the monic orthogonal polynomial-based methods are compared with those obtained by the 1stSDM. Finally, the monic orthogonal polynomial-based methods are proved to be the effective and efficient methods to predict the milling stability.

Orthogonal polynomials associated with an inverse spectral transform. The cubic case

The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn}n ? o defined by Qn(X) = Pn(X) + SnPn-1(X) + tnPn-2(X) + rnPn-3(X), n ? 1, where rn ? 0, n ? 3, and {Pn}n?0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters rn, sn, and tn can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.

A characterization of some semi classical orthogonal polynomials

We show that if is a sequence of monic orthogonal polynomials satisfying a differential-difference equation of the formthen the orthogonal polynomial sequence is necessarily orthogonal to a well known semi classical generalization of the classical Hermite weight. If is allowed to be any real number then we show that the Laguerre polynomials also appear as a limiting case. The method is based on a derivation of a pair of non linear difference equations which are a special case of the asymmetric discrete Painleve IV equations. Such equations appear entirely as a consequence of the differential-difference equation together with the three term recurrence relation for a general sequence of orthogonal polynomials as opposed to previous derivations that stem from the consideration of a specific weight function.

On the cubic decomposition of a family of Gq-semiclassical polynomial sequences of class one

In the present work, we deal with a monic orthogonal polynomial sequence (MOPS) inferred by its cubic decomposition . Our aim is to describe all -semiclassical q-polynomials of class one of these sequences. From the cubic decomposition structure and via the study of the q-difference equation satisfied by its corresponding regular form w where is the Hahn’s operator, only three MOPSs appear and their recurrence coefficients are explicitly given. A discrete measure representation of each corresponding linear functional is also established.

Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

For a natural extension of the circular unitary ensemble of order n, we study as \(n\rightarrow \infty \) the asymptotic behavior of the sequence of monic orthogonal polynomials \((\varPhi _{k,n}, k=0, \ldots , n)\) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, as \(n\rightarrow \infty \), the sequence of processes \((\log \varPhi _{\lfloor nt\rfloor ,n}(1), t \in [0,1])\) converges to a deterministic limit, and we describe the fluctuations and the large deviations.