Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

Let e ⊂ R be a finite union of + 1 disjoint closed intervals, and denote by ωj the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω1, . . . ,ω . Let {ãn, b̃n}∞n=−∞ be a point in the isospectral torus for e and p̃n its orthogonal polynomials. Let {an, bn}∞n=1 be a half-line Jacobi matrix with an = ãn + δan, bn = b̃n + δbn. Suppose ∞ ∑ n=1 |δan| + |δbn| <∞ and ∑N n=1 eδan, ∑N n=1 eδbn 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)/p̃n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J ’s for which this holds.