Asymptotic behavior of polynomials orthonormal on a homogeneous set

Abstract

Minor modifications are given to prove the Main Theorem under the Blaschke (instead of Carleson) condition as well as a small historical comment.

Cantor polynomials and some related classes of OPRL

“…. a n is the inverse of the leading coefficient of p n ), a result studied further in [1,20,21,10]. We conjecture:…”

“…Our model is the Szegő class of [21,10]. There is associated to any finite gap set, e, an isospectral torus of almost periodic Jacobi matrices, J, with σ ess (J) = e. The frequency module of the almost periodic functions is generated by the harmonic measures of parts of e between two points in the complement of e.…”

Two extensions of Lubinsky’s universality theorem

“…The point of this remark is that the construction in Section 2 relies on Jost solutions. For E's with rational harmonic measures, the Jacobi parameters are periodic; and Jost solutions can be constructed with Floquet theory rather than the more elaborate methods of [48,36,26,5].…”

“…We also note there that it suffices to prove the results in Section 2 when each interval has rational harmonic measure, so that one can use Floquet theory in place of the more subtle analysis of [48,36,26,5].…”

“…Our model is the measure associated to a point in the isospectral torus associated to E, where the analysis depends on results of Widom [48], Sodin-Yuditskii [36], Peherstorfer-Yuditskii [26], and Christiansen-SimonZinchenko [5].…”