Orthogonal polynomials and rational modifications of lebesgue measure on the unit circle. an inverse problem

Abstract

Let μ be a positive Borel measure on the unit circle with infinite support and φ_n(z)=φ_n(z,μ) be monic orthogonal polynomials with respect to μ. We give a full description of the class of measures possessing the following property: there exists a sequence of complex numbers (α_n) such that monic polynomials ψ_n=φ_n − α_nφ_n−1 are also orthogonal with respect to a certain measure v.

Keywords:

• Orthogonal polynomials
• Szegö functions
• positive measures

Classification Categories::

• 1991 Mathematics Subject Classification Primary 42C05