Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local asymptotic distribution of zeros of orthogonal polynomials

Authors: Vilmos Totik and Joseph L. Ullman
Journal: Trans. Amer. Math. Soc. 341 (1994), 881-894
MSC: Primary 42C05; Secondary 26C10
MathSciNet review: 1150019
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Converse results, which state a relation (inequality) for measures from that on their logarithmic potentials, are applied to local density of zeros of orthogonal polynomials when the measure of orthogonality is a general one with compact support. It will be shown that if the measure is sufficiently thick on a part of its support, then on that part the density of the zeros will be at least as large as the equilibrium measure of the support. A corresponding upper estimate on the distribution of the zeros will also be proved. All of our estimates are sharp, and they localize several well-known results.

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

  • [Be] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Philos. Soc. 41 (1945), 103–110. MR 0012325
  • [Br] Marcel Brelot, Sur l’allure des fonctions harmoniques et sousharmoniques à la frontière, Math. Nachr. 4 (1951), 298–307 (French). MR 0041989
  • [dVP1] Ch. J. de la Vallée-Poussin: Le potentiel logarithmique, Gauthier-Villars, Paris, 1949.
  • [dVP2] -, Potentiel et problème généralisé de Dirichlet, Math. Gaz. 22 (1938), 17-36.
  • [ErFr] P. Erdős and G. Freud, On orthogonal polynomials with regularly distributed zeros, Proc. London Math. Soc. (3) 29 (1974), 521–537. MR 0420119
  • [ErTu] Paul Erdös and Paul Turán, On interpolation. III. Interpolatory theory of polynomials, Ann. of Math. (2) 41 (1940), 510–553. MR 0001999
  • [Fr] G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971.
  • [Fu] B. Fuglede, Some properties of the Riesz charge associated with a $ \delta $-subharmonic function (manuscript).
  • [Gr] A. F. Grishin, Sets of regular growth of entire functions. I, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 40 (1983), 36–47 (Russian). MR 738442
  • [La] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • [StTo] Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828
  • [Ts] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894
  • [Ul1] J. L. Ullman, On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. (3) 24 (1972), 119–148. MR 0291718
  • [Ul2] Joseph L. Ullman, A survey of exterior asymptotics for orthogonal polynomials associated with a finite interval and a study of the case of the general weight measures, Approximation theory and spline functions (St. John’s, Nfld., 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 136, Reidel, Dordrecht, 1984, pp. 467–478. MR 786861
  • [Ul3] Joseph L. Ullman, Orthogonal polynomials for general measures. I, Rational approximation and interpolation (Tampa, Fla., 1983) Lecture Notes in Math., vol. 1105, Springer, Berlin, 1984, pp. 524–528. MR 783300, 10.1007/BFb0072438
  • [Ul4] Joseph L. Ullman, Orthogonal polynomials for general measures. II, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 247–254. MR 838990, 10.1007/BFb0076550
  • [Wi] Harold Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997–1013. MR 0209448
  • [Wy] M. F. Wyneken, The potential theory of Borel sets and some applications to the study of orthogonal polynomials (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42C05, 26C10

Retrieve articles in all journals with MSC: 42C05, 26C10

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society