This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The proposed project concerns perturbation theory of almost periodic Jacobi and CMV matrices with finite or infinite gap spectrum as well as asymptotic analysis of the associated orthogonal polynomials on the real line and the unit circle, respectively. The goal is to understand the relations between three fundamental objects: spectral measures (or measures of orthogonality), recursion coefficients, and orthogonal polynomials. The project investigates the interplay between regularity properties of the measures, asymptotic behavior of the coefficients, and asymptotics of the orthogonal polynomials. In particular, it extends what is known for the case of measures supported by a single interval to the case of measures supported by a union of several intervals.
Problems raised in this project bring together several areas of mathematics, most notably spectral theory and the theory of orthogonal polynomials. There are also important connections to harmonic analysis and the theory of Fuchsian groups. The proposed research will enhance our understanding of periodic and almost periodic structures with and without impurities. Potential areas of application span from random matrix theory to inverse problems in computer tomography and material science. The theory of orthogonal polynomials on the unit circle has also applications in geophysical scattering and electronic circuit filter design.
