Ladders operators for general discrete Sobolev orthogonal polynomials

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second--order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all the functions involved in these constructions can be computed explicitly

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

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

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

On asymptotic properties of Freud-Sobolev orthogonal polynomials

In this paper we consider a Sobolev inner product (f, g) S = fgd + # f # g # d (1) and we characterize the measures for which there exists an algebraic relation between the polynomials, orthogonal with respect to the measure and the polynomials, orthogonal with respect to (1), 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 = 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 ) and the Sobolev orthogonal polynomials Q n . Finally, we obtain some asymptotics for }

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

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

Manuel Alfaro, a,1 Juan J. Moreno-Balca ´ zar, b,c,2 a,,1,a

Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type I