Abstract
We provide leading-order asymptotics for the size of the gap in the zeros around 1 of paraorthogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (−1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Bandtlow, O.: Estimates for norms of resolvents and an application to the perturbation of spectra. Math. Nachr. 267, 3–11 (2004)
Davies, E.B., Simon, B.: Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle. J. Approx. Theory 141, 189–213 (2006)
Hislop, P.D., Sigal, I.M.: Introduction to Spectral Theory with Applications to Schrodinger Operators. Applied Mathematical Sciences. Springer, New York (1996)
Killip, R., Stoiciu, M.: Eigenvalue statistics for CMV matrices: From Poisson to Clock via random matrix ensembles. Duke Math. J. 146(3), 361–399 (2009)
Last, Y., Simon, B.: Fine structure of the zeros of orthogonal polynomials, IV. A priori bounds and clock behavior. Comm. Pure Appl. Math. 61, 486–538 (2008)
Nevai, P.: Orthogonal polynomials, measures and recurrences on the unit circle. Trans. Amer. Math. Soc. 300, 175–189 (1987)
Saff, E.B., Stylianopoulos, N.S.: Asymptotics for polynomial zeros: beware of predictions from plots. Comput. Methods Funct. Theory 8, 385–407 (2008)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part One: Classical Theory. Am. Math. Soc., Providence (2005)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Part Two: Spectral Theory. Am. Math. Soc., Providence (2005)
Simon, B.: Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures. Electron. Trans. Numer. Anal. 25, 328–368 (2006)
Simon, B.: Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle. J. Math. Anal. Appl. 329, 376–382 (2007)
Stoiciu, M.: The statistical distribution of the zeroes of random paraorthogonal polynomials on the unit circle. J. Approx. Theory 139, 29–64 (2006)
Wong, M.-W.L.: First and second kind paraorthogonal polynomials and their zeros. J. Approx. Theory 146, 282–293 (2007)
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Communicated by Serguei Denissov.
This research was partially supported by an NSF GRFP grant.
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Simanek, B. Zeros of Non-Baxter Paraorthogonal Polynomials on the Unit Circle. Constr Approx 35, 107–121 (2012). https://doi.org/10.1007/s00365-011-9127-x
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DOI: https://doi.org/10.1007/s00365-011-9127-x
Keywords
- Zeros of paraorthogonal polynomials
- Slow decay of Verblunsky coefficients
- CMV matrix
- Blaschke products
- Approximate eigenvectors