On rational transformations of linear functionals: direct problem

13 pages, no figures.-- MSC2000 codes: 33C47, 42C05.MR#: MR2086540 (2005f:42051)Zbl#: Zbl 1066.33008Let u be a quasi-definite linear functional. We find necessary and sufficient conditions in order to the linear functional v satisfying $(x-\tilde{a})u=\lambda(x-a)v$ be a quasi-definite one. Also we analyze some linear relations linking the polynomials orthogonal with respect to u and v.M.A. and M.L.R. were partially supported by MCYT Grant BFM 2003-06335-C03-03 (Spain), FEDER funds (EU), and DGA E-12/25 (Spain). F.M. was partially supported by MCYT Grant BFM 2003-06335-C03-02 (Spain) and INTAS Research Network NeCCA INTAS 03-51-66378. A.P. was partially supported by MCYT Grant BFM 2001-1793 (Spain), FEDER funds (EU), and DGA E-12/25 (Spain).Publicad

Some properties of zeros of Sobolev-type orthogonal polynomials

9 pages, no figures.-- MSC1991 code: 33C45.MR#: MR1391618 (97f:33008)Zbl#: Zbl 0862.33005For polynomials orthogonal with respect to a discrete Sobolev product, we prove that, for each n, Qn has at least n − m zeros on the convex hull of the support of the measure, where m denotes the number of terms in the discrete part. Interlacing properties of zeros are also described.Research by first (M.A.) and third (M.L.R.) authors was partially supported by Diputación General de Aragón P CB-12/91 and by Comisión Interministerial de Ciencia y Tecnología (CICYT-Spain) PB93-0228-C02-02.Publicad

Orthogonal polynomials associated with an inverse quadratic spectral transform

AbstractLet {Pn}n≥0 be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional u and {Qn}n≥0 a sequence of polynomials defined by Qn(x)=Pn(x)+snPn−1(x)+tnPn−2(x),n≥1, with tn≠0 for n≥2.We obtain a new characterization of the orthogonality of the sequence {Qn}n≥0 with respect to a linear functional v, in terms of the coefficients of a quadratic polynomial h such that h(x)v=u.We also study some cases in which the parameters sn and tn can be computed more easily, and give several examples.Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is presented

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

7 pages, no figures.-- MSC2000 codes: 33C45; 42C05.-- Issue title: "Special Functions, Information Theory, and Mathematical Physics" (Special issue dedicated to Professor Jesús Sánchez Dehesa on the occasion of his 60th birthday).Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,Qn(x) = Pn(x) + a1Pn-1(x) + ... + akPn-k(x), ak0, n>k.Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is shown.MA and MLR partially supported by Ministerio de Educación y Ciencia (MEC) of Spain under Grant MTM 2006-13000-C03-03 and Diputación General de Aragón (DGA) project E-64. FM partially supported by MEC of Spain under Grant MTM 2006-13000-C03-02 and INTAS Research Network NeCCA 03-51-6637. AP partially supported by MEC of Spain under Grants MTM 2004-03036 and MTM 2006-13000-C03-03 and DGA project E-64.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

Asymptotics for a generalization of Hermie polynomials

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin

On linearly related orthogonal polynomials and their functionals

13 pages, no figures.-- MSC2000 codes: 42C05.MR#: MR2010273 (2004i:33014)Zbl#: Zbl 1029.42014Let $\{P_n\}$ be a sequence of polynomials orthogonal with respect a linear functional $u$ and $\{Q_n\}$ a sequence of polynomials defined by $$P_n(x)+s_nP_{n-1}(x)=Q_n(x)+t_nQ_{n-1}(x).$$ We find necessary and sufficient conditions in order to $\{Q_n\}$ be a sequence of polynomials orthogonal with respect to a linear functional $v$. Furthermore, we prove that the relation between these linear functionals is $(x-\tilde a)u=\lambda (x-a)v$. Even more, if $u$ and $v$ are linked in this way we get that $\{P_n\}$ and $\{Q_n\}$ satisfy a formula as above.Second author (F.M.) was partially supported by Dirección General de Investigación MCYT Grant BFM 2003-06335-C03-02, Spain, and INTAS Project 2000-272. Third author (A.P.) was partially supported by Dirección General de Investigación MCYT Grant BFM 2001-1793, Spain. First and fourth authors (M.A. and M.L.R.) were partially supported by Dirección General de Investigación MCYT Grant BFM 2003-06335-C03-03, Spain, and Universidad de La Rioja, Spain.Publicad