Abstract
For Jacobi parameters belonging to one of 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.
Highlights
Let μ be a probability measure on the real line with infinite support such that for every n ∈ N0, the moments xn dμ(x) are finite
Let L2(R, μ) be the Hilbert space of square-integrable functions equipped with the scalar product f, g = f (x)g(x) dμ(x)
By performing the Gram–Schmidt orthogonalization process on the sequence of monomials one obtains the sequence of polynomials satisfying pn, pm = δnm where δnm is the Kronecker delta
Summary
Let μ be a probability measure on the real line with infinite support such that for every n ∈ N0, the moments xn dμ(x) are finite. The asymptotic behavior of Kn is well understood in the case when the measure μ has compact support In this setup one of the most general results has been proven in [29]. The measure μ has compact support, if and only if the Jacobi parameters are bounded. We study analogues of (1.6) and (1.7) for three different classes of Jacobi parameters: asymptotically periodic, periodically modulated and a blend of these two; for the definitions, see Sects. The third class has been studied in [1] as an example of unbounded Jacobi parameters corresponding to measures with absolutely continuous parts having supports equal a finite union of closed intervals. Theorem 1.2 Let (an) and (bn) be N -periodically modulated Jacobi parameters. C stands for a positive constant whose value may vary from occurrence to occurrence
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