Asymptotic zero distribution for a class of multiple orthogonal polynomials

Coussement, E.; Coussement, J.; Van Assche, W.

doi:10.1090/S0002-9947-08-04535-2

Asymptotic zero distribution for a class of multiple orthogonal polynomials
HTML articles powered by AMS MathViewer

by E. Coussement, J. Coussement and W. Van Assche PDF

Trans. Amer. Math. Soc. 360 (2008), 5571-5588 Request permission

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

M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1968.
A. I. Aptekarev, Multiple orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 423–447. MR 1662713, DOI 10.1016/S0377-0427(98)00175-7
Alexander I. Aptekarev, Pavel M. Bleher, and Arno B. J. Kuijlaars, Large $n$ limit of Gaussian random matrices with external source. II, Comm. Math. Phys. 259 (2005), no. 2, 367–389. MR 2172687, DOI 10.1007/s00220-005-1367-9
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), no. 1-2, 346–370. MR 2220045, DOI 10.1016/j.jat.2005.09.011
A. 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), no. 4, 487–524. MR 1771693, DOI 10.1007/s003650010004
Pavel M. Bleher and Arno B. J. Kuijlaars, Integral representations for multiple Hermite and multiple Laguerre polynomials, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 2001–2014 (English, with English and French summaries). MR 2187942
P. M. Bleher and A. B. J. Kuijlaars, Random matrices with external source and multiple orthogonal polynomials, Int. Math. Res. Not. 3 (2004), 109–129. MR 2038771, DOI 10.1155/S1073792804132194
Pavel Bleher and Arno B. J. Kuijlaars, Large $n$ limit of Gaussian random matrices with external source. I, Comm. Math. Phys. 252 (2004), no. 1-3, 43–76. MR 2103904, DOI 10.1007/s00220-004-1196-2
Marcel G. de Bruin, Simultaneous Padé approximation and orthogonality, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 74–83. MR 838972, DOI 10.1007/BFb0076532
Marcel G. de Bruin, Some aspects of simultaneous rational approximation, Numerical analysis and mathematical modelling, Banach Center Publ., vol. 24, PWN, Warsaw, 1990, pp. 51–84. MR 1097402
A. Bultheel, A. Cuyt, W. Van Assche, M. Van Barel, and B. Verdonk, Generalizations of orthogonal polynomials, J. Comput. Appl. Math. 179 (2005), no. 1-2, 57–95. MR 2134361, DOI 10.1016/j.cam.2004.09.036
Y. Ben Cheikh and N. Ben Romdhane. $d$-orthogonal polynomial sets of Chebyshev type. Preprint
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
K. Douak and P. Maroni, On $d$-orthogonal Tchebychev polynomials. I, Appl. Numer. Math. 24 (1997), no. 1, 23–53. MR 1454707, DOI 10.1016/S0168-9274(97)00006-8
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. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601, DOI 10.1090/chel/345
Jeffrey S. Geronimo and Theodore 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, DOI 10.1016/S0021-9045(02)00042-4
Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1991. Special functions—integral transforms—asymptotics—continued fractions; Reprint of the 1977 original; A Wiley-Interscience Publication. MR 1164865
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
A. B. J. Kuijlaars, Chebyshev quadrature for measures with a strong singularity, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 207–214. MR 1379132, DOI 10.1016/0377-0427(95)00110-7
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, DOI 10.1006/jath.2001.3617
A. B. J. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999), no. 1, 167–197. MR 1696553, DOI 10.1006/jath.1999.3316
K. Mahler, Perfect systems, Compositio Math. 19 (1968), 95–166 (1968). MR 239099
N. I. Muskhelishvili, Singular integral equations, Dover Publications, Inc., New York, 1992. 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. MR 1215485
Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. MR 1130396, DOI 10.1090/mmono/092
Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
Walter Van Assche, Multiple orthogonal polynomials, irrationality and transcendence, Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998) Contemp. Math., vol. 236, Amer. Math. Soc., Providence, RI, 1999, pp. 325–342. MR 1665377, DOI 10.1090/conm/236/03504
Walter Van Assche, Asymptotics for orthogonal polynomials and three-term recurrences, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 435–462. MR 1100305
Walter Van Assche and Els Coussement, Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001), no. 1-2, 317–347. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808581, DOI 10.1016/S0377-0427(00)00503-3
W. Van Assche and S. B. Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transform. Spec. Funct. 9 (2000), no. 3, 229–244. MR 1782974, DOI 10.1080/10652460008819257
A. Wintner. Spektraltheorie der Unendlichen Matrizen. Hirzel, Leipzig, 1929.