Communications in Mathematical Physics

, Volume 193, Issue 1, pp 125-150

First online:

Inverse Problem for Polynomials Orthogonal on the Unit Circle

  • J. S. GeronimoAffiliated withSchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-01660, USA.¶E-mail:
  • , R. JohnsonAffiliated withDipartimento Sistemi e informatica, Universita di Firenze, Firenze, Italy 50139.¶E-mail:

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Polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated. It is shown that the coefficients are in fact quasi-periodic. The measures associated with these quasi-periodic coefficients are exhibited and necessary and sufficient conditions relating quasi-periodicity and spectral measures of this type are given. Analogs for polynomials orthogonal on subsets of the real line are also presented.