On zeros of paraorthogonal polynomials
HTML articles powered by AMS MathViewer
- by Yen Chi Lun PDF
- Proc. Amer. Math. Soc. 147 (2019), 3389-3399 Request permission
Abstract:
We prove some results concerning the behaviour of zeros of families of paraorthogonal polynomials on the unit circle. We establish an interlacing property of the zeros of some functions related to the paraorthogonal polynomials. Monotonicity with respect to a parameter is also discussed in detail. A Markov type theorem is proved, and the monotonicity is also considered from a spectral point of view.References
- C. F. Bracciali, J. H. McCabe, T. E. Pérez, and A. Sri Ranga, A class of orthogonal functions given by a three term recurrence formula, Math. Comp. 85 (2016), no. 300, 1837–1859. MR 3471110, DOI 10.1090/mcom3041
- Cleonice F. Bracciali, Jairo S. Silva, A. Sri Ranga, and Daniel O. Veronese, Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences, Comput. Appl. Math. 37 (2018), no. 2, 1142–1161. MR 3804119, DOI 10.1007/s40314-016-0392-y
- Kenier Castillo, Monotonicity of zeros for a class of polynomials including hypergeometric polynomials, Appl. Math. Comput. 266 (2015), 183–193. MR 3377550, DOI 10.1016/j.amc.2015.05.058
- M. S. Costa, H. M. Felix, and A. Sri Ranga, Orthogonal polynomials on the unit circle and chain sequences, J. Approx. Theory 173 (2013), 14–32. MR 3073604, DOI 10.1016/j.jat.2013.04.009
- Dimitar K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx. Theory 85 (1996), no. 1, 88–97. MR 1382052, DOI 10.1006/jath.1996.0030
- Dimitar K. Dimitrov, Connection coefficients and zeros of orthogonal polynomials, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 331–340. MR 1858291, DOI 10.1016/S0377-0427(00)00653-1
- Dimitar K. Dimitrov, A late report on interlacing of zeros of polynomials, Constructive theory of functions, Prof. M. Drinov Acad. Publ. House, Sofia, 2012, pp. 69–79. MR 3051099
- Dimitar K. Dimitrov and Fernando R. Rafaeli, Monotonicity of zeros of Jacobi polynomials, J. Approx. Theory 149 (2007), no. 1, 15–29. MR 2371611, DOI 10.1016/j.jat.2007.04.004
- D. K. Dimitrov, M. E. H. Ismail, and A. Sri Ranga, A class of hypergeometric polynomials with zeros on the unit circle: extremal and orthogonal properties and quadrature formulas, Appl. Numer. Math. 65 (2013), 41–52. MR 3008187, DOI 10.1016/j.apnum.2012.11.002
- Dimitar K. Dimitrov and A. Sri Ranga, Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle, Math. Nachr. 286 (2013), no. 17-18, 1778–1791. MR 3145170, DOI 10.1002/mana.201200181
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
- Mourand E. H. Ismail and Xin Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992), no. 1, 131–140. MR 1079891, DOI 10.1090/S0002-9939-1992-1079891-5
- Mourad E. H. Ismail and David R. Masson, Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), no. 1, 1–40. MR 1354960, DOI 10.1006/jath.1995.1106
- Mourad E. H. Ismail and Martin E. Muldoon, A discrete approach to monotonicity of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 323 (1991), no. 1, 65–78. MR 1014251, DOI 10.1090/S0002-9947-1991-1014251-8
- M. E. H. Ismail and A. Sri Ranga, $R_{II}$ type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle, arXiv:1606.08055v4.
- M. E. H. Ismail and A. Sri Ranga, $R_{II}$ type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle, Linear Algebra Appl. 562 (2019), 63–90. MR 3886340, DOI 10.1016/j.laa.2018.10.005
- Mourad E. H. Ismail and Ruiming Zhang, On the Hellmann-Feynman theorem and the variation of zeros of certain special functions, Adv. in Appl. Math. 9 (1988), no. 4, 439–446. MR 968677, DOI 10.1016/0196-8858(88)90022-X
- André Markoff, Sur les racines de certaines équations, Math. Ann. 27 (1886), no. 2, 177–182 (French). MR 1510373, DOI 10.1007/BF01452056
- A. Martínez-Finkelshtein, A. Sri Ranga, and D. O. Veronese, Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure, Math. Comp. 87 (2018), no. 309, 261–288. MR 3716196, DOI 10.1090/mcom/3210
- Martin E. Muldoon, Properties of zeros of orthogonal polynomials and related functions, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 167–186. MR 1246857, DOI 10.1016/0377-0427(93)90321-2
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154
- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
- H. S. Wall and Marion Wetzel, Quadratic forms and convergence regions for continued fractions, Duke Math. J. 11 (1944), 89–102. MR 11340
Additional Information
- Yen Chi Lun
- Affiliation: DMAP, IBILCE, UNESP - Universidade Estadual Paulista, 15055-450 São José do Rio Preto, SP, Brazil
- MR Author ID: 1057491
- Email: yen.chilun@yahoo.com.tw
- Received by editor(s): April 3, 2018
- Received by editor(s) in revised form: August 14, 2018
- Published electronically: April 18, 2019
- Communicated by: Mourad Ismail
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3389-3399
- MSC (2010): Primary 30C45; Secondary 26C10
- DOI: https://doi.org/10.1090/proc/14359
- MathSciNet review: 3981117