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Applications of a new formula for OPUC with periodic Verblunsky coefficients

Abstract

We find a new formula for the orthonormal polynomials corresponding to a measure \(\mu \) on the unit circle whose Verblunsky coefficients are periodic. The formula is presented using the Chebyshev polynomials of the second kind and the discriminant of the periodic sequence. We present several applications including a resolution of a problem suggested by Simon (Commun Pure Appl Math 59(7):1042–1062, 2006) regarding the existence of singular points in the bands of the support of the measure and a universality result at all points of the essential support of \(\mu \).

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Notes

  1. Here and always, we identify the unit circle with the interval \([0,2\pi )\).

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Acknowledgements

The author would like to thank the anonymous referees for their careful reading of this manuscript and for their suggestions, especially for directing us to the references [4, 5, 9, 10, 13].

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Simanek, B. Applications of a new formula for OPUC with periodic Verblunsky coefficients. Res Math Sci 5, 42 (2018). https://doi.org/10.1007/s40687-018-0165-x

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  • DOI: https://doi.org/10.1007/s40687-018-0165-x

Keywords

  • Periodic Verblunsky coefficients
  • Chebyshev polynomials
  • Universality

Mathematics Subject Classification

  • Primary 42C05
  • Secondary 33C45