Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation I

Abstract

GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN A. For Jacobi parameters belonging to one of the three classes: asymptotically periodic, periodically modulated and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel-Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports. n j=0 p j (x)p j (y).To motivate the study of Christoffel-Darboux kernels see surveys [12] and [18].

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

“…Applications to Ignjatović's conjecture. In the following theorem we extend the results from[48, Section 4.3] and[50, Section 8.1] to the case when N = 1 and X 0 (0) is a non-trival parabolic element of SL(2, R). These results are motivated by[15, Conjecture 1].…”

“…Let us note that in (1.8) the exponent of a (j+1)N +i−1 is equal to 1 4 and it is different than in the cases I and IIa where it is equal to 1 2 (see [51, Theorem C] and [50,Theorem C]). Finally, we prove scaling limits of the Christoffel-Darboux kernel in the form analogous to [48,50].…”

“…The definition of ρ n reflects the unusual asymptotic behavior of the orthogonal polynomials. Indeed, in the cases I and IIa we have [48,Theorem C] and [50,Theorem D]). Also definitions of υ are different in all of the three cases.…”

About essential spectra of unbounded Jacobi matrices

“…Let us comment that in Theorem B, the absolute continuity of A follows by [41,Theorem B]. Moreover, by [39,Theorem 3.13] it stems that Λ is a union of N open disjoint bounded intervals. For r = 1 and under certain very strong assumptions, Theorem B has been proven in [1,Theorem 5].…”

“…The second class, that is blended Jacobi matrices (see Definition 2.4), has been introduced in [1] as an example of unbounded Jacobi matrices having absolutely continuous spectrum equal to a finite union of compact intervals. It has been further studied in [39,41] in the context of orthogonal polynomials.…”