#### Title

Finite Gap Jacobi Matrices, III. Beyond the Szego Class

#### 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

#### ISSN

0176-4276

#### Recommended Citation

"Finite Gap Jacobi Matrices, III. Beyond the Szego Class" (2012). *Faculty Bibliography 2010s*. 2423.

https://stars.library.ucf.edu/facultybib2010/2423

## Comments

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