Abstract
Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials Bn(.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials.
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References
M. Ambroladze, On exceptional sets of asymptotic relations for general orthogonal polynomials, J. Approx. Theory, 82 (1995), 257–273.
L. Daruis, P. González-Vera and O. Njåstad, Szegő quadrature formulas for certain Jacobi-type weight functions, submitted to Mathematics of Computation.
L. Ya. (aka Ya. L.) Geronimus, Orthogonal Polynomials, Consultants Bureau (New York, 1961).
Ya. L. Geronimus, Orthogonal Polynomials, English translation of the appendix to the Russian translation of Szegő's book [14], Fizmatgiz (Moscow, 1961), in: Two Papers on Special Functions, Amer. Math. Soc. Transl., series 2, Vol. 108 (Providence, Rhode Island, 1977), pp. 37–130.
L. Golinskii, Reflection coefficients for the generalized Jacobi weight functions, J. Approx. Theory, 78 (1994), 117–126.
L. Golinskii, Geronimus polynomials and weak convergence on a circular arc, Methods and applications of analysis, 6 (1999), 1–16.
U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, University of California Press (Berkeley, 1958), 2nd edition: Chelsea Publishing Company (New York, 1984).
W. B. Jones, O. Njåstad and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc., 21 (1989), 113–152.
S. Khrushchev, Schur's algorithm, orthogonal polynomials and convergence of Wall's continued fractions in L 2(T), to appear in J. Approx. Theory.
H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc., 16 (1987), 47–77.
P. Nevai, Orthogonal Polynomials, Memoirs of the Amer. Math. Soc., vol. 213 (1979).
E. B. Saff, Orthogonal polynomials from a complex perspective, in: Orthogonal Polynomials: Theory and Practice (P. Nevai, ed.), Kluwer Academic Publishers (Dordrecht, Boston, London, 1990), pp. 363–393.
E. B. Saff and V. Totik, What parts of a measure support attract zeros of the corresponding orthogonal polynomials?, Proc. Amer. Math. Soc., 114 (1992), 185–190.
G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc. (Providence, Rhode Island, 1975) (4th edition).
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Golinski, L. Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. Acta Mathematica Hungarica 96, 169–186 (2002). https://doi.org/10.1023/A:1019765002077
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DOI: https://doi.org/10.1023/A:1019765002077