Algunos resultados recientes en polinomios ortogonales de Sobolev

En este trabajo expondré algunos resultados recientes acerca de polinomios ortogonales con respecto a un producto escalar no estándar que involucra medidas de soporte no acotado

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L(α,M,N) n (x) orthogonal with respect to the Sobolev inner product (p, q) = 1 Γ(α+1) Z ∞ 0 p(x)q(x) x α e −x dx + M p(0)q(0) + N p 0 (0)q 0 (0), N,M ≥ 0, α > −1, firstly introduced by Koekoek and Meijer in 1993 and extensively studied in the last years. We present some new asymptotic properties of these polynomials and also a limit relation between the zeros of these polynomials and the zeros of Bessel function Jα(x). The results are illustrated with numerical examples. Also, some general asymptotic formulas for generalizations of these polynomials are conjectured.Junta de AndalucíaDirección General de InvestigaciónUnión Europe

Mehler-Heine asymptotics of a class of generalized hypergeometric polynomials

We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures

Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials

6 pages, no figures.-- MSC1991 codes: 33C25; 42CO5.Zbl#: Zbl 0895.33003We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f,g)_S= + λ where = \int_{-1}\sp 1f(x)g(x)(1-x\sp 2)\sp {\alpha-\frac{1}{2}}dx, with α > -1/2 and λ > 0. The asymptotics of the zeros and norms of these polynomials are also established.Research by first (A.M.F.) and second (J.J.M.B.) was partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad

k-Coherence of measures with non-classical weights

7 pages, no figures.-- MSC2000 codes: Primary 42C05; Secondary 33C25.MR#: MR1882617 (2003b:42042)The concept of k-coherence of two positive measures μ1 and μ2 is useful in the study of the Sobolev orthogonal polynomials. If μ1 or μ2 are compactly supported on R then any 0-coherent pair or symmetrically 1-coherent pair (μ1, μ2) must contain a Jacobi measure (up to affine transformation). Here examples of k-coherent pairs (k ≥ 1) when neither μ1 nor μ2 are Jacobi are constructed.Research of F. Marcellán supported by Dirección General de Investigación(Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01. Research of A. Martínez-Finkelshtein partially supported by INTAS project 2000-272, a research grant of Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, and by Junta de Andalucía, Grupo de Investigación FQM 0229. Research of J. J. Moreno-Balcázar partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229, and INTAS project 2000-272.Publicad

Asymptotic behavior of varying discrete Jacobi-Sobolev orthogonal polynomials

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently. (C) 2016 Elsevier B.V. All rights reserved.The authors JFMM and JJMB are partially supported by Research Group FQM-0229 (belonging to Campus of International Excellence CEIMAR). The author JFMM is funded by a grant of Plan Propio de la Universidad de Almería. The author FM is partially supported by Dirección General de Investigación, Ministerio de Economía y Competitividad Innovación of Spain, Grant MTM2012-36732-C03-01. The author JJMB is partially supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain and European Regional Development Found, grants MTM2011-28952-C02-01 and MTM2014-53963-P, and Junta de Andalucía (excellence grant P11-FQM-7276)

Inner products involving q-differences: the little q-Laguerre-Sobolev polynomials

22 pages, no figures.-- MSC codes: Primary 33C25; Secondary 33D45.-- Issue title: "Higher transcendental functions and their applications".MR#: MR1765938 (2001d:33018)Zbl#: Zbl 0957.33008In this paper, polynomials which are orthogonal with respect to the inner product \multline\langle p,r\rangle_S=\sum^\infty_{k=0}p(q^k)r(q^k) {(aq)^k(aq;q)_\infty\over(q;q)_k}\\ +\lambda\sum^\infty_{k=0} (D_qp)(q^k)(D_qr)(q^k){(aq)^k(aq;q)_\infty\over(q;q)_k},\endmultline where $D_q$ is the $q$-difference operator, $\lambda\geq0,\ 0<q<1$ and $0<aq<1$, are studied. For these polynomials, algebraic properties and $q$-difference equations are obtained as well as their relation with the monic little $q$-Laguerre polynomials. Some properties of the zeros of these polynomials are also deduced. Finally, the relative asymptotics $\{Q_n(x)/p_n(x;a )\}_n$ on compact subsets of {\bf C}\sbs[0,1] is given, where $Q_n(x)$ is the $n$th degree monic orthogonal polynomial with respect to the above inner product and $p_n(x;a )$ denotes the monic little $q$-Laguerre polynomial of degree $n$.E.G. wishes to acknowledge partial financial support by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB-96-0952. The research of F.M. was partially supported by DGES of Spain under Grant PB96-0120-C03-01 and INTAS Project 93-0219 Ext. J.J.M.B. also wishes to acknowledge partial financial support by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad

Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

13 pages, no figures.-- MSC codes: 42C05; 33C25; 39A10.MR#: MR1741786 (2000k:42032)Zbl#: Zbl 0984.42016We study the analytic properties of the monic Meixner-Sobolev polynomials $\{Q_n\}$ orthogonal with respect to the inner product involving differences $$(p,q)_S=\sum infty_{i=0}[p(i)q(i)+\lambda\Delta p(i)\Delta q(i)] {\mu (\gamma)_i\over i!},$$ $\gamma>0,\ 0<\mu<1$, where $\lambda\geq0,\ \Delta$ is the forward difference operator $(\Delta f(x)=f(x+1)-f(x))$ and $(\gamma)_n$ denotes the Pochhammer symbol. Relative asymptotics for Meixner-Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner-Sobolev polynomials and Plancherel-Rotach asymptotics for $\{Q_n\}$ are deduced.The work of E.G. has been partially supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB-96-0952. The work of F.M. is partially supported by PB96-0120-C03-01 and INTAS-93-0219 Ext. The work of J.J.M.-B. is partially supported by Junta de Andalucía, G.I. FQM0229.Publicad

Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

16 pages, no figures.-- MSC2000 codes: 42C05.-- Issue title: "Proceedings of the Seventh International Symposium on Orthogonal Polynomials, Special Functions and Applications" (University of Copenhagen, Denmark, Aug 18-22, 2003).MR#: MR2127867 (2006a:33005)Zbl#: Zbl 1060.42015Let $\{Q_n(x)\}_n$ be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product $$\bigl\langle (p(x),r(x)\bigr \rangle_S=\bigl\langle{\bold u}_0,p(x)r(x) \bigr\rangle+ \lambda\bigl\langle {\bold u}_1,(\Delta p)(x)(\Delta r)(x) \bigr\rangle,$$ where $\lambda\ge 0$, $(\Delta f)(x)=f(x+1)-f(x)$ denotes the forward difference operator and $({\bold u}_0,{\bold u}_1)$ is a $\Delta$-coherent pair of positive-definite linear functionals being ${\bold u}_1$ the Meixner linear functional. In this paper, relative asymptotics for the $\{Q_n(x)\}_n$ sequence with respect to Meixner polynomials on compact subsets of \bbfC\setminus[0,+\infty) is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-$\Delta$-coherent pair, that is, when ${\bold u}_0={\bold u}_1$ is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.The work by I.A. and E.G. was partially supported by Ministerio de Ciencia y Tecnología of Spain under grant BFM2002-04314-C02-01. The work by F.M. has been supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2003-06335-C03-02 as well as by the NATO collaborative grant PST.CLG. 979738. The work by J.J.M.B has been supported by Dirección General de Investigación of Spain under grant BFM2001-3878-C02-02 as well as by Junta de Andalucía (research group FQM0229).Publicad

Sobolev orthogonal polynomials: balance and asymptotics

Let μ0 and μ1 be measures supported on an unbounded interval and Sn,λn the extremal varying Sobolev polynomial which minimizes _\lambda_n=\int P^2 d\mu_0+\lambda_n \int P'^2 d\mu_1, \lambda_n>0 in the class of all monic polynomials of degree n. The goal of this paper is twofold. On one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence (λn) such that both measures μ0 and μ1 play a role in the asymptotics of (Sn,λn). On the other, we apply such ideas to the case when both μ0 and μ1 are Freud weights. Asymptotics for the corresponding Sn,λn are computed, illustrating the accuracy of the choice of λn