Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

Abstract

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.MSC: Primary 42C05; Secondary 33C45

Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives

“…The case j = 1 has attracted the interest of many researchers from many points of view. First, the analysis of the asymptotic properties of such orthogonal polynomials is done (see [1] and the survey paper [10]). Second, the study of analytic properties of their zeros such as interlacing with the zeros of Laguerre polynomials and monotonicity of each zero in terms of N, among others, are completely discussed in [4].…”

“…Second, the study of analytic properties of their zeros such as interlacing with the zeros of Laguerre polynomials and monotonicity of each zero in terms of N, among others, are completely discussed in [4]. Furthermore, the limit behavior of such zeros is also deduced in [10]. Third, the study of spectral properties of these orthogonal polynomials in the sense that they are eigenfunctions of an infinite order linear differential operator which reduces to order 2α + 8 if α is a nonnegative integer number and N > 0 is presented in [9].…”

Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

“…Hence, (14) means that ϕ + (b k ) is a zero of the polynomial 2N j=0 A j z j of multiplicity at least N k + 1 − I k .…”

“…[9,10,16,[21][22][23]25]) nor from the point of view of Sobolev orthogonality, where it remains like a partially explored subject [3]. In fact, new results continue to appear in some recent publications [4,[6][7][8][11][12][13][14][15][18][19][20].…”

“…The study of the asymptotic properties of the sequence (B n ) ∞ n=0 has been analyzed considering the cases 'mass points inside' or 'mass points outside' of supp μ, respectively, being supp μ a compact subset of R or, more recently, an unbounded interval of the real line (see, for instance [7,14]). The first results in the literature about asymptotic properties for orthogonal polynomials with respect to a Sobolev type inner product like (1) appear in [15], where the authors studied the case when there is only one mass point supporting the derivatives either inside or outside [−1, 1] and μ is a measure in the Nevai class M(0, 1).…”