Orthogonal polynomials on the unit circle and chain sequences

• Autores: M.S. Costa, H.M. Felix, A. Sri Ranga
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 173, Nº 1, 2013, págs. 14-32
• Idioma: inglés
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• Resumen

  • Szeg½o has shown that real orthogonal polynomials on the unit circle can be mapped to orthogonal polynomials on the interval [-1, 1] by the transformation 2x = z + z-1. In the 80�s and 90�s Delsarte and Genin showed that real orthogonal polynomials on the unit circle can be mapped to symmetric orthogonal polynomials on the interval [-1, 1] using the transformation 2x = z1/2 + z-1/2. We extend the results of Delsarte and Genin to all orthogonal polynomials on the unit circle. The transformation maps to functions on [-1, 1] that can be seen as extensions of symmetric orthogonal polynomials on [-1, 1] satisfying a threeterm recurrence formula with real coefficients {cn} and {dn}, where {dn} is also a positive chain sequence.

    Via the results established, we obtain a characterization for a point w (|w| = 1) to be a pure point of the measure involved. We also give a characterization for orthogonal polynomials on the unit circle in terms of the two sequences {cn} and {dn}.