On Some Classes of Polynomials Orthogonal on Arcs of the Unit Circle Connected with Symmetric Orthogonal Polynomials on an Interval

Starting from the Delsarte?Genin (DG) mapping of the symmetric orthogonal polynomials on an interval (OPI) we construct a one-parameter family of polynomials orthogonal on the unit circle (OPC). The value of the parameter defines the arc on the circle where the weight function vanishes. Some explicit examples of OPC connected with generic Askey?Wilson polynomials are constructed. These polynomials can be considered, as a four-parameter extension of the Askey?Szego? polynomials on the unit circle. We also present examples of OPC having finite and infinite purely discrete spectrum and addition masses inside and outside the unit circle. Connections of DG transformation with sieved OPI and OPC, chain sequences, and the Bauer's numericalg-algorithm (in approximation theory) are analyzed. In particular, we construct some classes of “semiclassical” OPI and OPC starting from periodic solutions of the Bauer'sg-algorithm.

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