18 research outputs found
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
and 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=â« f(x)g(x)dÎŒ1(x)+Ï â« fâČ(x)gâČ(x)dÎŒ2(x), Ï > 0, with x outside of the support of the measure ÎŒ2. We assume that ÎŒ1 and ÎŒ2 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 ăf,gă=â«â11f(x)g(x)(1âx2)αâ12dx+â«fâČ(x)gâČ(x)dÏ(x),α>â12, where dÏ is a measure involving a Gegenbauer weight and with mass points outside the interval (â1,1). 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 MehlerâHeine type formula. These results are illustrated with some numerical experiments
Asymptotics for JacobiâSobolev orthogonal polynomials associated with non-coherent pairs of measures
AbstractWe consider the Sobolev inner product ăf,gă=â«â11f(x)g(x)dÏ(α,ÎČ)(x)+â«fâČ(x)gâČ(x)dÏ(x), where dÏ(α,ÎČ)(x)=(1âx)α(1+x)ÎČdx with α,ÎČ>â1, and Ï is a measure involving a rational modification of a Jacobi weight and with a mass point outside the interval (â1,1). 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 MehlerâHeine 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 FreudâSobolev 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 , and we characterize the measures ÎŒ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure ÎŒ 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 supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight )and the Sobolev orthogonal polynomials Qn. Finally, we obtain some asymptotics for {Qn}.Research by first author (A.C.) partially supported by DirecciĂłn General de InvestigaciĂłn (Ministerio de Ciencia y TecnologĂa) of Spain under Grant BFM 2000 0015. Research by second author (F.M.) partially supported by DirecciĂłn General de InvestigaciĂłn (Ministerio de Ciencia
y TecnologĂa) 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 AndalucĂa, Grupo de InvestigaciĂłn FQM 0229, DirecciĂłn General de InvestigaciĂłn (Ministerio de Ciencia y TecnologĂa) 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 JacobiâSobolev 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 ÂŽ zar, b,c,2 a,,1,a
LaguerreâSobolev orthogonal polynomials: asymptotics for coherent pairs of type I