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Inequalities for the Zeros of an Orthogonal Expansion of a Polynomial

  • G. Schmeisser
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 430)

Abstract

Turán pointed out the importance of studying the location of the zeros of a polynomial in terms of the coefficients of an orthogonal expansion. He himself obtained numerous results for the Hermite expansion. Later Specht showed in a series of papers that analogous theorems hold for any expansion with respect to a system of polynomials orthogonal on the real line. His work stimulated various further studies. We give a survey on this topic with special emphasis on some results from an unpublished manuscript of Specht and new contributions by the author.

Key words and phrases

Inequalities Zeros Bound of zeros Orthogonal Expansion Algebraic polynomials Orthogonal polynomials Norm estimates 

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References

  1. [1]
    S. Barnett, A companion matrix analogue for orthogonal polynomials, Linear Algebra Appl. 12 (1975), 197–208.zbMATHCrossRefGoogle Scholar
  2. [2]
    F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math. 2 (1960), 137–141.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    A. T. Bharucha-Reid and M. Sambandham, Random Polynomials, Academic Press, Orlando, 1986.zbMATHGoogle Scholar
  4. [4]
    T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon & Breach, New York, 1978.zbMATHGoogle Scholar
  5. [5]
    A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. 32 (1995), 1–37.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    J. Favard, Sur les polynômes de Tchebycheff, C.R. Acad. Sci. Paris 200 (1935), 2052–2053.Google Scholar
  7. [7]
    A. Giroux, Estimates of the imaginary parts of the zeros of a polynomial, Proc. Amer. Math. Soc. 44 (1974), 61–67.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    E. M. Gol’berg and V. N. Malozemov, Estimates for the zeros of certain polynomials, Vestnik Leningrad Univ. Math. 6 (1979), 127-135 [Transl, from Vestnik Leningrad Mat. Mekh. Astronom. (1973), No. 7, 18-24].Google Scholar
  9. [9]
    Lajos László, Imaginary part bounds on polynomial zeros, Linear Algebra Appl. 44 (1982), 173–180.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    E. Makai and P. Turán, Hermite expansion and distribution of zeros of polynomials, Publ. Math. Inst. Hung. Acad. Sci. Ser. A 8 (1963), 157–163.zbMATHGoogle Scholar
  11. [11]
    M. Marden, Geometry of Polynomials, Math. Surveys 3, Amer. Math. Soc., Providence, R.I., 1966.Google Scholar
  12. [12]
    G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publ., Singapore — New Jersey — London — Hong Kong, 1994.zbMATHGoogle Scholar
  13. [13]
    D. S. Mitrinović, Analytic Inequalities, Springer Verlag, Berlin — Heidelberg — New York, 1970.zbMATHGoogle Scholar
  14. [14]
    N. Obreschkoff, Über die Wurzeln von algebraischen Gleichungen, Jahresber. Deutsch. Math.-Verein. 33 (1924), 52–64.zbMATHGoogle Scholar
  15. [15]
    Q. I. Rahman and G. Schmeisser, forthcoming book on polynomials, to be edited by Oxford University Press.Google Scholar
  16. [16]
    G. Schmeisser, Optimale Schranken zu einem Satz über Nullstellen Hermitescher Trinôme, J. Reine Angew. Math. 246 (1971), 147–160.MathSciNetzbMATHGoogle Scholar
  17. [17]
    G. Schmeisser, Nullstelleneinschlie²ungen und Landau-Fejér-Montel Problem, Studia Sci. Math. Hung. 7 (1972), 459–472.MathSciNetGoogle Scholar
  18. [18]
    W. Specht, Die Lage der Nullstellen eines Polynoms, Math. Nachr. 15 (1956), 353–374.MathSciNetCrossRefGoogle Scholar
  19. [19]
    W. Specht, Die Lage der Nullstellen eines Polynoms, II, Math. Nachr. 16 (1957), 257–263.MathSciNetCrossRefGoogle Scholar
  20. [20]
    W. Specht, Die Lage der Nullstellen eines Polynoms, III, Math. Nachr. 16 (1957), 369–389.MathSciNetCrossRefGoogle Scholar
  21. [21]
    W. Specht, Die Lage der Nullstellen eines Polynoms, IV, Math. Nachr. 21 (1960), 201–222.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    ——, Zur Analysis der Polynome, unpublished typed manuscript written not later than 1964.Google Scholar
  23. [23]
    G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, 4th edn., Providence, R.I., 1975.Google Scholar
  24. [24]
    P. Turán, Sur l’algèbre fonctionelle, Comptes Rendus du Premier Congr. Math. Hongr. 1950, Akad. Kiadó, Budapest, 1952, pp. 279-290.Google Scholar
  25. [25]
    P. Turán, H ermite-expansion and strips for zeros of polynomials, Arch. Math. 5 (1954), 148–152.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    P. Turán, To the analytic theory of algebraic equations, Izvestija Mat. Inst. Bulg. Akad. Nauk. 3 (1959), 123–137.Google Scholar
  27. [27]
    R. Vermes, On the zeros of a linear combination of polynomials, Pacific J. Math. 19 (1966), 553–559.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • G. Schmeisser
    • 1
  1. 1.Mathematisches Institut der Universität Erlangen-NürnbergErlangenGermany

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