|
Asymptotic zero distribution for a class of multiple orthogonal polynomials
Author(s):
E.
Coussement;
J.
Coussement;
W.
Van Assche
Journal:
Trans. Amer. Math. Soc.
360
(2008),
5571-5588.
MSC (2000):
Primary 33C45, 42C05;
Secondary 15A18
Posted:
May 20, 2008
MathSciNet review:
2415086
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Piñeiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.
References:
-
- 1.
- M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1968.
- 2.
- A. I. Aptekarev. Multiple orthogonal polynomials. J. Comput. Appl. Math. 99 (1998), 423-447. MR 1662713 (99m:42036)
- 3.
- A. I. Aptekarev, P. M. Bleher, and A. B. J. Kuijlaars. Large
 limit of Gaussian random matrices with external source, part II. Comm. Math. Phys. 259 (2005), 367-389. MR 2172687 (2006f:82036) - 4.
- A. I. Aptekarev, V. Kalyagin, G. López Lagomasino, and I. A. Rocha. On the limit behavior of recurrence coefficients for multiple orthogonal polynomials. J. Approx. Theory 139 (2006), 346-370. MR 2220045 (2007a:42048)
- 5.
- A. I. Aptekarev, V. Kaliaguine, and J. Van Iseghem. The genetic sums' representation for the moments of a system of Stieltjes functions and its application. Constr. Approx. 16 (2000), 487-524. MR 1771693 (2001g:41021)
- 6.
- P. M. Bleher and A. B. J. Kuijlaars. Integral representations for multiple Hermite and multiple Laguerre polynomials. Annales de l'Institut Fourier 55 (2005), 2001-2014. MR 2187942 (2007c:33014)
- 7.
- P. M. Bleher and A. B. J. Kuijlaars. Random matrices with external source and multiple orthogonal polynomials. Internat. Math. Research Notices 2004:3 (2004), 109-129. MR 2038771 (2004k:82034)
- 8.
- P. M. Bleher and A. B. J. Kuijlaars. Large
 limit of Gaussian random matrices with external source, part I. Comm. Math. Phys. 252 (2004), 43-76. MR 2103904 (2005i:82031) - 9.
- M. G. de Bruin. Simultaneous Padé approximation and orthogonality. In Orthogonal polynomials and applications (C. Brezinski et al., ed.), volume 1171 of Lecture Notes in Mathematics, pages 74-83. Springer, Berlin, 1985. MR 838972 (87m:41019)
- 10.
- M. G. de Bruin. Some aspects of simultaneous rational approximation. In Numerical Analysis and Mathematical Modeling, volume 24 of Banach Center Publications, pages 51-84. PWN-Polish Scientific Publishers, Warsaw, 1990. MR 1097402 (92b:41030)
- 11.
- A. Bultheel, A. Cuyt, M. Van Barel, B. Verdonk, and W. Van Assche. Generalizations of orthogonal polynomials. J. Comput. Appl. Math. 179 (2005), 57-95. MR 2134361 (2006a:42034)
- 12.
- Y. Ben Cheikh and N. Ben Romdhane.
 -orthogonal polynomial sets of Chebyshev type. Preprint - 13.
- T. S. Chihara. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
- 14.
- K. Douak and P. Maroni. On
 -orthogonal Tchebychev polynomials, I. Appl. Num. Math. 24 (1997), 23-53. MR 1454707 (98g:42038) - 15.
- F. P. Gantmacher and M. G. Krein. Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Revised edition. AMS Chelsea Publishing, Providence, RI, 2002. MR 1908601 (2003f:34161)
- 16.
- J. Geronimo and T. P. Hill. Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform. J. Approx. Theory 121 (2003), no. 1, 54-60. MR 1962995 (2004a:60040)
- 17.
- P. Henrici. Special functions - integral transforms - asymptotics - continued fractions, volume 2 of Applied and computational complex analysis (reprint of the 1977 original). Wiley Classics Library, Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. MR 1164865 (93b:30001)
- 18.
- M. E. H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable, volume 98 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 2005. MR 2191786 (2007f:33001)
- 19.
- A. B. J. Kuijlaars. Chebyshev quadrature for measures with strong singularity. J. Comp. Appl. Math. 65 (1995), 207-214. MR 1379132 (96m:41050)
- 20.
- A. B. J. Kuijlaars and S. Serra Capizzano. Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients. J. Approx. Theory 113 (2001), no. 1, 142-155. MR 1866252 (2002i:42029)
- 21.
- A. B. J. Kuijlaars and W. Van Assche. The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Appr. Theory 99 (1999), 167-197. MR 1696553 (2000h:42015)
- 22.
- K. Mahler. Perfect systems. Compositio Math. 19 (1968), 95-166. MR 0239099 (39:458)
- 23.
- N. I. Muskhelishvili. Singular integral equations, Boundary problems of function theory and their application to mathematical physics, Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok. Corrected reprint of the 1953 English translation, Dover Publications, Inc., New York, 1992. MR 1215485 (94a:45001)
- 24.
- P. Nevai. Orthogonal polynomials. Mem. Amer. Math. Soc. 213 (1979). MR 519926 (80k:42025)
- 25.
- E. M. Nikishin and V. N. Sorokin. Rational Approximants and Orthogonality, volume 92 in Translations of Mathematical Monographs. Amer. Math. Soc., Providence, RI, 1991. MR 1130396 (92i:30037)
- 26.
- E. B. Saff and V. Totik. Logarithmic Potentials with External Fields, volume 316 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin/NY, 1997. MR 1485778 (99h:31001)
- 27.
- W. Van Assche. Multiple orthogonal polynomials, irrationality and transcendence. In volume 236 of Continued fractions: from Analytic Number Theory to Constructive Approximation, Contemporary Mathematics (B. C. Berndt et al., ed.), pages 325-342. Amer. Math. Soc., Providence RI, 1999. MR 1665377 (2000k:42039)
- 28.
- W. Van Assche. Asymptotics for orthogonal polynomials and three-term recurrences. In ``Orthogonal Polynomials'' (P. Nevai, ed.), volume 294 of NATO ASI Series C, pages 435-462. Kluwer Academic, Dordrecht, 1990. MR 1100305 (92k:42038)
- 29.
- W. Van Assche and E. Coussement. Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127 (2001), 317-347. MR 1808581 (2001i:33012)
- 30.
- W. Van Assche and S. B. Yakubovich. Multiple orthogonal polynomials associated with Macdonald functions. Integral Transforms Special Funct. 9 (2000), 229-244. MR 1782974 (2001e:42034)
- 31.
- A. Wintner. Spektraltheorie der Unendlichen Matrizen. Hirzel, Leipzig, 1929.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
33C45, 42C05,
15A18
Retrieve articles in all Journals with
MSC (2000):
33C45, 42C05,
15A18
Additional Information:
E.
Coussement
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
J.
Coussement
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
W.
Van Assche
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
Email:
walter@wis.kuleuven.be
DOI:
10.1090/S0002-9947-08-04535-2
PII:
S 0002-9947(08)04535-2
Keywords:
Multiple orthogonal polynomials,
asymptotics
Received by editor(s):
June 19, 2006
Received by editor(s) in revised form:
January 31, 2007
Posted:
May 20, 2008
Additional Notes:
This work was supported by INTAS project 03-51-6637, by FWO projects G.0455.04 and G.0184.02 and by OT/04/21 of Katholieke Universiteit Leuven
The second author is a postdoctoral researcher at the Katholieke Universiteit Leuven (Belgium)
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
