Asymptotics for Jacobi–Sobolev orthogonal polynomials associated with non-coherent pairs of measures

AbstractWe consider the Sobolev inner product 〈f,g〉=∫−11f(x)g(x)dψ(α,β)(x)+∫f′(x)g′(x)dψ(x), where dψ(α,β)(x)=(1−x)α(1+x)βdx with α,β>−1, and ψ is a measure involving a rational modification of a Jacobi weight and with a mass point outside the interval (−1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product on different regions of the complex plane. In fact, we obtain the outer and inner strong asymptotics for these polynomials as well as the Mehler–Heine asymptotics which allow us to obtain the asymptotics of the largest zeros of these polynomials. We also show that in a certain sense the above inner product is also equilibrated

Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.Comment: Changed content; 34 pages, 41 reference

Sobolev orthogonal polynomials and spectral methods in boundary value problems

The work by FM has been supported by FEDER/Ministerio de Ciencia e Innovación-Agencia Estatal de Investigación of Spain, grant PID2021-122154NB-I00, and the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors, grant EPUC3M23 in the context of the V PRICIT (Regional Program of Research and Technological Innovation). LF, TEP and MAP thanks Grant FQM-246-UGR20 funded by Consejería de Universidad, Investigación e Innovación and by European Union NextGenerationEU/PRTR; and Grant CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. Funding for APC: Universidad Carlos III de Madrid (Agreement CRUE-Madroño 2023).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.FEDER/Ministerio de Ciencia e Innovación-Agencia Estatal de Investigación of Spain PID2021-122154NB-I00Madrid Government EPUC3M23Consejería de Universidad, Investigación e Innovación FQM-246-UGR20European Union NextGenerationEU/PRTRMCIN/AEI/10.13039/501100011033: CEX2020-001105-MUniversidad Carlos III de Madrid CRUE-Madroño 202

On asymptotic properties of Freud–Sobolev orthogonal polynomials

16 pages, no figures.-- MSC2000 codes: 33C45; 33C47; 42C05.MR#: MR2016838 (2005e:33004)Zbl#: Zbl 1043.33005In this paper we consider a Sobolev inner product $(f,g)_S=\int fg\,d\mu+ \lambda \int f'g'\,d\mu (*)$, and we characterize the measures μ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure μ and the polynomials, {Qn}, orthogonal with respect to (*), such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a Freud weight. In particular, we study the case $d\mu=e^{-x^4}dx$ supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight $e^{-x^4}$)and the Sobolev orthogonal polynomials Qn. Finally, we obtain some asymptotics for {Qn}.Research by first author (A.C.) partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM 2000 0015. Research by second author (F.M.) partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM2003 06335 C03 02, by INTAS Project 2000 272 and by the NATO collaborative Grant PST.CLG. 979738. Research by third author (J.J.M.-B.) partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229, Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM 2001 3878 C02 02 and INTAS Project 2000 272.Publicad

Asymptotics of Sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support

AbstractWe study the strong asymptotics for the sequence of monic polynomials Qn(x), orthogonal with respect to the inner product (f,g)s=∫ f(x)g(x)dμ1(x)+σ ∫ f′(x)g′(x)dμ2(x), σ > 0, with x outside of the support of the measure μ2. We assume that μ1 and μ2 are symmetric and compactly supported measures on R satisfying a coherence condition. As a consequence, the asymptotic behaviour of (Qn,Qn)s and of the zeros of Qn is obtained

Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

AbstractWe consider the Sobolev inner product 〈f,g〉=∫−11f(x)g(x)(1−x2)α−12dx+∫f′(x)g′(x)dψ(x),α>−12, where dψ is a measure involving a Gegenbauer weight and with mass points outside the interval (−1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler–Heine type formula. These results are illustrated with some numerical experiments

On Sobolev orthogonal polynomials

Sobolev orthogonal polynomials have been studied extensively in the past quarter-century. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.The project was carried out when the second author was on sabbatical from the University of Oregon and visited the Carlos III University of Madrid under its generous Excellence Chair Program. The work of the first author has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01. The work of the second author was supported in part by National Science Foundation Grant DMS-1106113

Coherent pairs of measures and Markov-Bernstein inequalities

All the coherent pairs of measures associated to linear functionals $c_0$ and $c_1$, introduced by Iserles et al in 1991, have been given by Meijer in 1997. There exist seven kinds of coherent pairs. All these cases are explored in order to give three term recurrence relations satisfied by polynomials. The smallest zero $\mu_{1,n}$ of each of them of degree $n$ has a link with the Markov-Bernstein constant $M_n$ appearing in the following Markov-Bernstein inequalities: $$c_1((p^\prime)^2) \le M_n^2 c_0(p^2), \quad \forall p \in \mathcal{P}_n,$$ where $M_n=\frac{1}{\sqrt{\mu_{1,n}}}$. The seven kinds of three term recurrence relations are given. In the case where $c_0 =e^{-x} dx+\delta(0)$ and $c_1 =e^{-x} dx$, explicit upper and lower bounds are given for $\mu_{1,n}$, and the asymptotic behavior of the corresponding Markov-Bernstein constant is stated. Except in a part of one case, $\lim_{n \to \infty} \mu_{1,n}=0$ is proved in all the cases.Comment: 32 page

Δ-Coherent pairs and orthogonal polynomials of a discrete variable

27 pages, no figures.-- MSC1991 codes: 33C25; 42C05.MR#: MR1949214 (2003m:33009)Zbl#: Zbl 1047.42019In this paper we define the concept of D-coherent pair of linear functionals. We prove that if (u_0, u_1) is a D-coherent pair of linear functionals then at least one of them must be a classical discrete linear functional under certain conditions. Examples related to Meixner and Hahn linear functionals are given.F. Marcellán wishes to acknowledge Dirección General de Investigación (MCYT) of Spain for financial support under grant BFM2000-0206C04-01 and INTAS project 2000-272.Publicad