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Finite gap Jacobi matrices. III: Beyond the Szegő class. (English) Zbl 1238.42009
Summary:
[Parts I and II in ibid. 32, No. 1, 1–65 (2010; Zbl 1200.42012) and ibid. 33, No. 3, 365–403 (2011; Zbl 1236.42021)]
Let $${\mathfrak{e}} \subset {\mathbb{R}}$$ be a finite union of $$\ell+1$$ disjoint closed intervals, and denote by $$\omega_{j}$$ the harmonic measure of the $$j$$ left-most bands. The frequency module for $${\mathfrak{e}}$$ is the set of all integral combinations of $$\omega_{1},\dots,\omega_{\ell}$$. Let $$\{{\tilde{a}}_{n},{\tilde{b}}_{n}\}_{n=-\infty}^{\infty}$$ be a point in the isospectral torus for $${\mathfrak{e}}$$ and $$\tilde{p}_{n}$$ its orthogonal polynomials. Let $$\{a_{n},b_{n}\}_{n=1}^{\infty}$$ be a half-line Jacobi matrix with $$a_{n} = {\tilde{a}}_{n} + \delta a_{n}$$, $$b_{n} = {\tilde{b}}_{n} +\delta b_{n}$$. Suppose $\sum_{n=1}^\infty | \delta a_n| ^2 + | \delta b_n| ^2 <\infty$ and $$\sum_{n=1}^{N} e^{2\pi i\omega n} \delta a_{n}$$, $$\sum_{n=1}^{N} e^{2\pi i\omega n} \delta b_{n}$$ have finite limits as $$N \to \infty$$ for all $$\omega$$ in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to $$\omega$$, then for $$z \in \mathbb{C} \setminus \mathbb{R}$$, $$p_{n}(z)/{\tilde{p}}_{n}(z)$$ has a limit as $$n \to \infty$$. Moreover, we show that there are non-Szegő class $$J$$’s for which this holds.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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##### References:
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