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Remarks on the $ (C,-1)$-summability of the distribution of zeros of orthogonal polynomials

Authors: Paul Nevai and Walter Van Assche
Journal: Proc. Amer. Math. Soc. 122 (1994), 759-767
MSC: Primary 42C05
MathSciNet review: 1204382
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Abstract: Given $ {x_1} < {x_2} < \cdots < {x_n}$ and $ {y_1} < {y_2} < \cdots < {y_{n - 1}}$, two interlacing sequences of real numbers, the rectangular diagram for these numbers is a continuous piecewise linear function with slopes $ \pm 1$ and with n local minima at the points $ {x_i}$ and $ n - 1$ local maxima at the points $ {y_j}$. Recently, S. Kerov determined the asymptotic behavior of the rectangular diagrams associated with the zeros of two consecutive orthogonal polynomials for which the coefficients in the three-term recurrence relation converge. The purpose of this note is to show how this result of S. Kerov and even some of its generalizations follow directly from certain $ (C, - 1)$-summability results on distribution of zeros of orthogonal polynomials proved by us some time ago.

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  • [1] E. W. Cheney, Introduction to approximation theory, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1982) edition. MR 1656150
  • [2] S. V. Kerov, Generalized Hall-Littlewood symmetric functions and orthogonal polynomials, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 67–94. MR 1166196
  • [3] -, The asymptotics of interlacing sequences and the growth of continual Young diagrams, preprint, 1992.
  • [4] -, The asymptotics for interlacing roots of orthogonal polynomials, Algebra i Analiz (1993) (to appear). (Russian)
  • [5] S. V. Kerov, Transition probabilities of continual Young diagrams and the Markov moment problem, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32–49, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 104–117. MR 1251166, 10.1007/BF01085981
  • [6] Paul G. Nevai, Distribution of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 249 (1979), no. 2, 341–361. MR 525677, 10.1090/S0002-9947-1979-0525677-5
  • [7] Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, 10.1090/memo/0213
  • [8] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
  • [9] Walter Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987. MR 903848
  • [10] -, Orthogonal polynomials on non-compact sets, Acad. Analecta, Meded. Konink. Acad. Wetensch. Lett. Sch. Kunsten België 51 (1989), Nr. 2, 1-36.
  • [11] 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

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Keywords: Orthogonal polynomials, zeros, rectangular diagrams
Article copyright: © Copyright 1994 American Mathematical Society