Finite Gap Jacobi Matrices, III. Beyond the Szego Class

    Authors

    J. S. Christiansen; B. Simon;M. Zinchenko

    Comments

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    Abbreviated Journal Title

    Constr. Approx.

    Keywords

    Szego asymptotics; Orthogonal polynomials; Almost periodic sequences; Slowly decaying perturbations; PERTURBATIONS; BOUNDS; Mathematics

    Abstract

    Let e. R be a finite union of l + 1 disjoint closed intervals, and denote by omega(j) the harmonic measure of the j left- most bands. The frequency module for e is the set of all integral combinations of omega(1), ... , omega(l). Let {(a)over tilde (n), (b)over tilde (n)}(n=-8)(infinity) be a point in the isospectral torus for e and (p)over tilde(n) its orthogonal polynomials. Let {a(n), b(n)}(n=1)(infinity) be a half-line Jacobi matrix with a(n) = a(n) + delta(an), b(n) = (b)over tilde(n) + delta b(n). Suppose [GRAPHICS] and Sigma(n)(n=1) e(2 pi i omega N) delta a(n), Sigma(N)(n=1) e2pi.ndbn have finite limits as N -> 8 for all. in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to omega, then for z is an element of C \ R, p(n)(z)/(p) over tilde n(z) has a limit as n - >infinity. Moreover, we show that there are non-Szego class J's for which this holds.

    Journal Title

    Constructive Approximation

    Volume

    35

    Issue/Number

    2

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    259

    Last Page

    272

    WOS Identifier

    WOS:000300521500006

    ISSN

    0176-4276

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