Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

Abstract

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e −ϕ(x) , giving a unified treatment for the so-called Freud (i.e., when ϕ has polynomial growth at infinity) and Erdös (when ϕ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

“…In this sense, in [2] it is obtained an upper bound for the distance of the zeros to the convex hull of the support, under certain conditions on the measures. For an updated review on the analytic properties of Sobolev polynomials orthogonal with respect to exponential type weights supported on unbounded sets of the real line we recommend [9] and for a more condensed reviewing the introduction of [8] and [1]. The results obtained here are the analogues of situations studied in [8] and [1], for the case where the support is the real semiaxis, with the natural aided difficulties.…”

“…As a consequence, we obtain the logarithmic (n-root) asymptotics for the Sobolev orthogonal polynomials for exponential weights. To this end, a result about asymptotically extremal polynomials with respect to varying weights, established by the authors in a previous paper (see [1,Theorem 1.1]), plays a crucial role. Along with it, we give a new, and simpler, proof of the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support, previously obtained by Durán and Saff [2].…”

“…In [1] a similar analysis is carried out, but for Sobolev inner products with exponential weights supported in the whole real axis. In order to extend those results to the case where the support is the semiaxis it is possible to follow a similar approach, but some modifications are needed.…”

“…Indeed, let Σ be a closed set and ω an admissible weight on Σ. In [1], the authors considered asymptotically extremal polynomials {p n } with respect to varying weights {ω n } n∈N , that is, when it holds…”

Asymptotic behaviours and general recurrence relations

“…In the literature, many fine properties of this type of polynomials can be found, for example in [2] and [8], . .…”