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˜n,b˜n}∞n=−∞ be a point in the isospectral torus for e and p˜n its orthogonal polynomials. Let {an,bn}∞n=1 be a half-line Jacobi matrix with an=a˜n+δan , bn=b˜n+δbn . Suppose
∑n=1∞∣δan∣2+∣δbn∣2<∞
and ∑Nn=1e2πiωnδan , ∑Nn=1e2πiωnδbn 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, pn(z)/p˜n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds.