Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II (original) (raw)

Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

Numerical Algorithms, 2012

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product leftlanglep,qrightrangleS=int0inftyp(x)q(x)xalphae−xdx+Npprime(a)qprime(a),alpha>−1\left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1leftlanglep,qrightrangleS=int0inftyp(x)q(x)xalphae−xdx+Npprime(a)qprime(a),alpha>−1 where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler–Heine formula as well as a Plancherel–Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N.

A new approach to the asymptotics for Sobolev orthogonal polynomials

arXiv (Cornell University), 2010

In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to (f, g) = f gdµ + r i=0 M i f (i) (0)g (i) (0), M i ≥ 0, where µ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to µ, Mehler-Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these results we use a new approach different from the methods used in the literature up to now. The development of this technique is highly motivated by the fact that the methods used when µ is bounded do not work.

A new approach to the asymptotics of Sobolev type orthogonal polynomials

Journal of Approximation Theory, 2011

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points. c

Zeros of varying Laguerre–Krall orthogonal polynomials

Proceedings of the American Mathematical Society, 2013

In this paper we introduce a sequence of varying orthogonal polynomials related to a Laguerre weight where this absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the origin. The main objective is to obtain asymptotic relations between the zeros of these polynomials and the zeros of the Bessel functions of the first kind (or linear combinations of them). This is done through Mehler–Heine type formulas. With these relations we can easily compute asymptotically the zeros of these polynomials. We show some numerical experiments.

Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs

Journal of Computational and Applied Mathematics, 2001

We study the strong asymptotics for the sequence of manic polynomials Q&c), orthogonal with respect to the inner product U-3 9)s = s f(xMx) h(x) + 1 s f'(x)s'(x> 44X), A> 0, with x outside of the support of the measure ~2. We assume that ~1 and ~2 are symmetric and compactly supported measures on lR satisfying a coherence condition. As a consequence, the asymptotic behaviour of (Q",Q")s and of the zeros of Qn is obtained.

Asymptotics for varying discrete Sobolev orthogonal polynomials

Applied Mathematics and Computation, 2017

We consider a varying discrete Sobolev inner product such as (f, g) S = f (x) g(x) dμ + M n f (j) (c) g (j) (c) , where μ is a finite positive Borel measure supported on an infinite subset of the real line, c is adequately located on the real axis, j ≥0, and { M n } n ≥0 is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler-Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c , just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler-Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones.

The Diagonal General Case of the Laguerre-Sobolev Type Orthogonal Polynomials

Revista Colombiana de Matemáticas, 2013

We consider the family of polynomials orthogonal with respect to the Sobolev type inner product corresponding to the diagonal general case of the Laguerre-Sobolev type orthogonal polynomials. We analyze some properties of these polynomials, such as the holonomic equation that they satisfy and, as an application, an electrostatic interpretation of their zeros. We also obtain a representation of such polynomials as a hypergeometric function, and study the behavior of their zeros.