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Applications of a new formula for OPUC with periodic Verblunsky coefficients

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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 \([0,2\pi )\).

References

  1. Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer, New York (1995)

    MATH  Google Scholar 

  2. Bourgade, P.: On Random Matrices and \(L\)-Functions, Ph.D. Thesis, New York University. http://www.cims.nyu.edu/~bourgade/papers/PhDThesis.pdf

  3. Cantero, M., Iserles, A.: From orthogonal polynomials on the unit circle to functional equations via generating functions. Trans. Am. Math. Soc. 368(6), 4027–4063 (2016)

    Article  MathSciNet  Google Scholar 

  4. Charris, J., Ismail, M.E.H.: Sieved orthogonal polynomials VII: generalized polynomial mappings. Trans. Am. Math. Soc. 340(1), 71–93 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Charris, J., Ismail, M.E.H., Monsalve, S.: On sieved orthogonal polynomials X: general blocks of recurrence relations. Pac. J. Math. 163(2), 237–267 (1994)

    Article  MathSciNet  Google Scholar 

  6. Danka, T.: Universality limits for generalized Jacobi measures. Adv. Math. 316, 613–666 (2017)

    Article  MathSciNet  Google Scholar 

  7. Danka, T., Totik, V.: Christoffel functions with power type weights. J. Eur. Math. Soc. 20(3), 747–796 (2018)

    Article  MathSciNet  Google Scholar 

  8. de Jesus, M.N., Petronilho, J.: On orthogonal polynomials obtained via polynomial mappings. J. Approx. Theory 162(12), 2243–2277 (2010)

    Article  MathSciNet  Google Scholar 

  9. Geronimo, J., Van Assche, W.: Orthogonal polynomials with asymptotically periodic recurrence coefficients. J. Approx. Theory 46(3), 251–283 (1986)

    Article  MathSciNet  Google Scholar 

  10. Geronimo, J., Van Assche, W.: Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Am. Math. Soc. 308(2), 559–581 (1988)

    Article  MathSciNet  Google Scholar 

  11. Geronimous, Y.: Sur quelques équations aux différences finies et les systèmes correspondants des polynômes orthogonaux. C. R. (Doklady) Acad. Sci. URSS (N.S.) 29, 536–538 (1940)

    MathSciNet  MATH  Google Scholar 

  12. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  13. Ismail, M.E.H., Li, X.: On sieved orthogonal polynomials IX: orthogonality on the unit circle. Pac. J. Math. 153(2), 289–297 (1992)

    Article  MathSciNet  Google Scholar 

  14. Korenev, B.G.: Bessel Functions and Their Applications, Translated from the Russian by E. V. Pankratiev. Analytical Methods and Special Functions 8, Taylor & Francis, Ltd., London (2002)

  15. Levin, E., Lubinsky, D.: Universality limits involving orthogonal polynomials on the unit circle. Comput. Methods Funct. Theory 7, 543–561 (2007)

    Article  MathSciNet  Google Scholar 

  16. Lubinsky, D.: Universality limits at the hard edge of the spectrum for measures with compact support. Int. Math. Res. Not. IMRN 2008 (2008)

  17. Lubinsky, D., Nguyen, V.: Universality limits involving orthogonal polynomials on an arc of the unit circle. Comput. Methods Funct. Theory 13(1), 91–106 (2013)

    Article  MathSciNet  Google Scholar 

  18. Lukashov, A.: Circular parameters of polynomials that are orthogonal on several arcs of the unit circle. Mat. Sb. 195(11), 95–118 (2004)

    Article  MathSciNet  Google Scholar 

  19. Máté, A., Nevai, P., Totik, V.: Szegő’s extremum problem on the unit circle. Ann. Math. (2) 134(2), 433–453 (1991)

    Article  MathSciNet  Google Scholar 

  20. McLaughlin, J.: Combinatorial identities deriving from the \(n\)th power of a \(2\times 2\) matrix. Integers: Electr. J. Combin. Number Theory 4, A19 (2004)

    MATH  Google Scholar 

  21. Peherstorfer, F., Steinbauer, R.: Orthogonal polynomials on arcs of the unit circle. II. Orthogonal polynomials with periodic reflection coefficients. J. Approx. Theory 87(1), 60–102 (1996)

    Article  MathSciNet  Google Scholar 

  22. Peherstorfer, F., Steinbauer, R.: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. J. Approx. Theory 88(3), 316–353 (1997)

    Article  MathSciNet  Google Scholar 

  23. Peherstorfer, F., Steinbauer, R.: Strong asymptotics of orthonormal polynomials with the aid of Green’s function. SIAM J. Math. Anal. 32(2), 385–402 (2000)

    Article  MathSciNet  Google Scholar 

  24. Peherstorfer, F., Steinbauer, R.: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II. Weak asymptotics. J. Approx. Theory 105(1), 102–128 (2000)

    Article  MathSciNet  Google Scholar 

  25. Peherstorfer, F., Steinbauer, R.: Orthogonal polynomials on the circumference and arcs of the circumference. J. Approx. Theory 102(1), 96–119 (2000)

    Article  MathSciNet  Google Scholar 

  26. Pintér, F., Nevai, P.: Schur functions and orthogonal polynomials on the unit circle, in “Approximation Theory and Function Series”. Bolyai Soc. Math. Stud. 5, 293–306 (1996). (János Bolyai Math. Soc., Budapest)

    MATH  Google Scholar 

  27. Simanek, B.: The Bergman shift operator on polynomial lemniscates. Constr. Approx. 41(1), 113–131 (2015)

    Article  MathSciNet  Google Scholar 

  28. Simanek, B.: Two universality results for polynomial reproducing kernels. J. Approx. Theory 216, 16–37 (2017)

    Article  MathSciNet  Google Scholar 

  29. Simanek, B.: Universality at an endpoint for orthogonal polynomials with Geronimus-type weights. Proc. Am. Math. Soc. 146(9), 3995–4007 (2018)

    Article  MathSciNet  Google Scholar 

  30. Simon, B.: Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line. J. Approx. Theory 126(2), 198–217 (2004)

    Article  MathSciNet  Google Scholar 

  31. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part One: Classical Theory. American Mathematical Society, Providence, RI (2005)

    MATH  Google Scholar 

  32. Simon, B.: Orthogonal Polynomials on the Unit Circle, Part Two: Spectral Theory. American Mathematical Society, Providence, RI (2005)

    MATH  Google Scholar 

  33. Simon, B.: Fine structure of the zeros of orthogonal polynomials. III. Periodic recursion coefficients. Commun. Pure Appl. Math. 59(7), 1042–1062 (2006)

    Article  MathSciNet  Google Scholar 

  34. Simon, B.: The Christoffel-Darboux kernel, in Perspectives in partial differential equations, harmonic analysis and applications. In: Proceedings of Symposia in Pure Mathematics, vol. 79, pp. 295–335 American Mathematical Society, Providence, RI (2008)

  35. Simon, B.: Szegő’s Theorem and Its Descendants, Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials. Princeton University Press, Princeton, NJ (2011)

    MATH  Google Scholar 

  36. Stahl, H., Totik, V.: General Orthogonal Polynomials. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  37. Totik, V.: Universality and fine zero spacing on general sets. Ark. Mat. 47(2), 361–391 (2009)

    Article  MathSciNet  Google Scholar 

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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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  1. Baylor University, Waco, TX, USA

    Brian Simanek

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  1. Brian Simanek
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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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