Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

Let e ⊂ R be a finite union of ℓ+1 disjoint closed intervals, and denote by ω_j the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω_1,…,ω_ℓ. Let {a_nb_n}^∞_(n=−∞) be a point in the isospectral torus for e and p_n its orthogonal polynomials. Let {a_nb_n}^∞_(n=1) be a half-line Jacobi matrix with a_n=a_n+δa_n, b_n=b_n+δb_n. Suppose ∑^∞_(n=1)│δan│^2 + │δb_n│^2 < ∞ and ∑^N_n=1^e^(2πiωn), δa_n ∑^N_n=1^e^(2πiωn) δb_n have finite limits as N → ∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, p_n(z)p_n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J's for which this holds.