Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle


Author: A. Sri Ranga
Journal: Proc. Amer. Math. Soc. 144 (2016), 1129-1143
MSC (2010): Primary 42C05, 33C47; Secondary 33C45
DOI: https://doi.org/10.1090/proc12766
Published electronically: July 1, 2015
MathSciNet review: 3447666
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The principal objective here is to look at some algebraic properties of the orthogonal polynomials $ \Psi _n^{(b,s,t)}$ with respect to the Sobolev inner product on the unit circle

$\displaystyle \langle f,g\rangle _{S^{(b,s,t)}} = (1-t)\, \langle f,g\rangle _{... ...ine {f(1)}\,g(1) + s\, \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}}, $

where $ \langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\o... ...heta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta . $ Here, $ \mathcal {R}e(b) > -1/2$, $ 0 \leq t < 1$, $ s > 0$ and $ \tau (b)$ is taken to be such that $ \langle 1,1\rangle _{\mu ^{(b)}} = 1$. We show that, for example, the monic Sobolev orthogonal polynomials $ \Psi _n^{(b,s,t)}$ satisfy the recurrence $ \Psi _n^{(b,s,t)}(z) - \beta _n^{(b,s,t)} \Psi _{n-1}^{(b,s,t)}(z) = \Phi _n^{(b,t)}(z), $ $ n \geq 1$, where $ \Phi _n^{(b,t)}$ are the monic orthogonal polynomials with respect to the inner product $ \langle f,g\rangle _{\mu ^{(b,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1)$. Some related bounds and asymptotic properties are also given.

References [Enhancements On Off] (What's this?)

  • [1] A. Aptekarev, E. Berriochoa, and A. Cachafeiro, Strong asymptotics for the continuous Sobolev orthogonal polynomials on the unit circle, J. Approx. Theory 100 (1999), no. 2, 381-391. MR 1715020 (2000i:42015), https://doi.org/10.1006/jath.1999.3358
  • [2] Gábor Szegő, Collected papers. Vol. 1, Contemporary Mathematicians, Birkhäuser, Boston, Mass., 1982. 1915-1927; Edited by Richard Askey; Including commentaries and reviews by George Pólya, P. C. Rosenbloom, Askey, L. E. Payne, T. Kailath and Barry M. McCoy. MR 674482 (84d:01082a)
  • [3] E. Berriochoa and A. Cachafeiro, A family of Sobolev orthogonal polynomials on the unit circle, J. Comput. Appl. Math. 105 (1999), no. 1-2, 163-173. MR 1690584 (2000h:42013), https://doi.org/10.1016/S0377-0427(99)00040-0
  • [4] A. C. Berti, C. F. Bracciali, and A. Sri Ranga, Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials, Numer. Algorithms 34 (2003), no. 2-4, 203-216. MR 2043896 (2005c:42024), https://doi.org/10.1023/B:NUMA.0000005363.32764.d3
  • [5] Andrea C. Berti and A. Sri Ranga, Companion orthogonal polynomials: some applications, Appl. Numer. Math. 39 (2001), no. 2, 127-149. MR 1862329 (2002k:33006), https://doi.org/10.1016/S0168-9274(01)00046-0
  • [6] A. Branquinho, A. Foulquié Moreno, F. Marcellán, and M. N. Rebocho, Coherent pairs of linear functionals on the unit circle, J. Approx. Theory 153 (2008), no. 1, 122-137. MR 2432558 (2009e:42048), https://doi.org/10.1016/j.jat.2008.03.003
  • [7] M. J. Cantero, L. Moral, and L. Velázquez, Measures and para-orthogonal polynomials on the unit circle, East J. Approx. 8 (2002), no. 4, 447-464. MR 1952510 (2004i:42021)
  • [8] K. Castillo, M. S. Costa, A. Sri Ranga, and D. O. Veronese, A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula, J. Approx. Theory 184 (2014), 146-162. MR 3218796, https://doi.org/10.1016/j.jat.2014.05.007
  • [9] K. Castillo, L. E. Garza, and F. Marcellán, Zeros of Sobolev orthogonal polynomials on the unit circle, Numer. Algorithms 60 (2012), no. 4, 669-681. MR 2946488, https://doi.org/10.1007/s11075-012-9594-6
  • [10] Kenier Castillo, Luis E. Garza, and Francisco Marcellán, Asymptotic behaviour of Sobolev orthogonal polynomials on the unit circle, Integral Transforms Spec. Funct. 24 (2013), no. 1, 23-38. MR 3011920, https://doi.org/10.1080/10652469.2011.649751
  • [11] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884 (58 #1979)
  • [12] 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, https://doi.org/10.1016/j.jat.2013.04.009
  • [13] Leyla Daruis, Olav Njåstad, and Walter Van Assche, Para-orthogonal polynomials in frequency analysis, Rocky Mountain J. Math. 33 (2003), no. 2, 629-645. MR 2021369 (2004j:42019), https://doi.org/10.1216/rmjm/1181069970
  • [14] M. G. de Bruin, W. G. M. Groenevelt, and H. G. Meijer, Zeros of Sobolev orthogonal polynomials of Hermite type, Appl. Math. Comput. 132 (2002), no. 1, 135-166. MR 1918711 (2004f:33016), https://doi.org/10.1016/S0096-3003(01)00183-7
  • [15] Antonia M. Delgado and Francisco Marcellán, Companion linear functionals and Sobolev inner products: a case study, Methods Appl. Anal. 11 (2004), no. 2, 237-266. MR 2143522 (2006f:42022)
  • [16] 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, https://doi.org/10.1002/mana.201200181
  • [17] G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.
  • [18] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79. MR 0061706 (15,869i)
  • [19] L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), no. 3, 169-186. MR 1919160 (2003e:41048), https://doi.org/10.1023/A:1019765002077
  • [20] W. G. M. Groenevelt, Zeros of Sobolev orthogonal polynomials of Gegenbauer type, J. Approx. Theory 114 (2002), no. 1, 115-140. MR 1880298 (2002m:42025), https://doi.org/10.1006/jath.2001.3643
  • [21] A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), no. 2, 151-175. MR 1104157 (92b:42029), https://doi.org/10.1016/0021-9045(91)90100-O
  • [22] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, With two chapters by Walter Van Assche; With a foreword by Richard A. Askey, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. MR 2191786 (2007f:33001)
  • [23] M. N. de Jesus and J. Petronilho, Sobolev orthogonal polynomials and $ (M,N)$-coherent pairs of measures, J. Comput. Appl. Math. 237 (2013), no. 1, 83-101. MR 2966889, https://doi.org/10.1016/j.cam.2012.07.006
  • [24] William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113-152. MR 976057 (90e:42027), https://doi.org/10.1112/blms/21.2.113
  • [25] D. H. Kim, K. H. Kwon, F. Marcellán, and G. J. Yoon, Zeros of Jacobi-Sobolev orthogonal polynomials, Int. Math. J. 4 (2003), no. 5, 413-422. MR 2027148 (2004m:33017)
  • [26] Francisco Marcellán, Teresa E. Pérez, and Miguel A. Piñar, Gegenbauer-Sobolev orthogonal polynomials, Nonlinear numerical methods and rational approximation, II (Wilrijk, 1993), Math. Appl., vol. 296, Kluwer Acad. Publ., Dordrecht, 1994, pp. 71-82. MR 1307190 (95m:42032)
  • [27] H. G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), no. 3, 321-343. MR 1451509 (99c:42046), https://doi.org/10.1006/jath.1996.3062
  • [28] H. G. Meijer and M. G. de Bruin, Zeros of Sobolev orthogonal polynomials following from coherent pairs, J. Comput. Appl. Math. 139 (2002), no. 2, 253-274. MR 1876639 (2002k:42053), https://doi.org/10.1016/S0377-0427(01)00421-6
  • [29] Ana Foulquié Moreno, Francisco Marcellán, and K. Pan, Asymptotic behavior of Sobolev-type orthogonal polynomials on the unit circle, J. Approx. Theory 100 (1999), no. 2, 345-363. MR 1714949 (2000k:42035), https://doi.org/10.1006/jath.1999.3357
  • [30] Barry Simon, Orthogonal polynomials on the unit circle. Part 1, Classical theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. MR 2105088 (2006a:42002a)
  • [31] Barry Simon, Orthogonal polynomials on the unit circle. Part 2, Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. MR 2105089 (2006a:42002b)
  • [32] A. Sri Ranga, Szegő polynomials from hypergeometric functions, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4259-4270. MR 2680052 (2011j:42052), https://doi.org/10.1090/S0002-9939-2010-10592-0
  • [33] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517 (51 #8724)
  • [34] Walter Van Assche, Orthogonal polynomials in the complex plane and on the real line, Special functions, $ q$-series and related topics (Toronto, ON, 1995), Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 211-245. MR 1448688 (98i:33014)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42C05, 33C47, 33C45

Retrieve articles in all journals with MSC (2010): 42C05, 33C47, 33C45


Additional Information

A. Sri Ranga
Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP - Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
Email: ranga@ibilce.unesp.br

DOI: https://doi.org/10.1090/proc12766
Keywords: Orthogonal polynomials on the unit circle, Sobolev orthogonal polynomials on the unit circle, para-orthogonal polynomials, positive chain sequences
Received by editor(s): October 31, 2014
Received by editor(s) in revised form: November 1, 2014, January 28, 2015, and February 13, 2015
Published electronically: July 1, 2015
Additional Notes: This work received support from the funding bodies CNPq (grant No. 475502/2013-2) and FAPESP (Grant No. 2009/13832-9) of Brazil
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society