Advertisement

Bernstein theorems for harmonic functions

  • Thomas Bagby
  • Norman Levenberg
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1550)

Keywords

Harmonic Function Compact Subset Holomorphic Function Open Neighborhood Extremal Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. H. Armitage, T. Bagby, and P. M. Gauthier, Note on the decay of elliptic equations, Bull. London Math. Soc. 17 (1985), pp. 554–556.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    D. H. Armitage and M. Goldstein, Better than uniform approximation on closed sets by harmonic functions with singularities, Proc. London Math. Soc. (4) 60 (1990), pp. 319–343.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T. Bagby, Approximation in the mean by solutions of elliptic equations, Trans. Amer. Math. Soc. 281 (1984), pp. 761–784.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. S. Baouendi and C. Goulaouic, Approximation of analytic functions on compact sets and Bernstein's inequality, Trans. Amer. Math. Soc. 189 (1974), pp. 251–261.MathSciNetzbMATHGoogle Scholar
  5. [5]
    T. Bloom, On the convergence of multivariate Lagrange interpolants, Constructive Approximation 5 (1989), pp. 415–435.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J.-P. Demailly, Measures de Monge-Ampère et mesures plurisousharmoniques, Math. Zeitschrift 194 (1987), pp. 519–564.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Dineen, The Schwarz lemma, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1989.zbMATHGoogle Scholar
  8. [8]
    R. Harvey and J. C. Polking, A Laurent expansion for solutions to elliptic equations, Trans. Amer. Math. Soc. 180 (1973), pp. 407–413.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), pp. 231–240.MathSciNetzbMATHGoogle Scholar
  10. [10]
    M. Klimek, Pluripotential theory, to appear.Google Scholar
  11. [11]
    Nguyen Thanh Van and B. Djebbar, Propriétés asymptotiques d'une suite orthonormale de polynomes harmoniques, Bull. Sc. math. (5) 113 (1989), pp. 239–251.zbMATHGoogle Scholar
  12. [12]
    W. Plesniak, On some polynomial conditions of the type of Leja in Cn, in Analytic functions Kozubnik 1979, Proceedings, Lecture Notes in Mathematics vol. 798, Springer-Verlag, pp. 384–391.Google Scholar
  13. [13]
    J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), pp. 322–357.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Siciak, Asymptotic behavior of harmonic polynomials bounded on a compact set, Ann. Polon. Math. 20 (1968), pp. 267–278.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J. Siciak, Extremal plurisubharmonic functions in CN, Ann. Polon. Math. 39 (1981), pp. 175–211.MathSciNetzbMATHGoogle Scholar
  16. [16]
    J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Coll. Publ. vol. 20, Third Edition, 1960.Google Scholar
  17. [17]
    J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bull. Amer. Math. Soc. 35 (1929), pp. 499–544.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. L. Walsh, Maximal convergence of sequences of harmonic polynomials, Ann. of Math. 38 (1937), pp. 321–364.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    V.P. Zaharjuta, Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. 33 (1976), pp. 137–148.MathSciNetGoogle Scholar

Copyright information

© The Euler International Mathematical Institute 1993

Authors and Affiliations

  • Thomas Bagby
    • 1
    • 2
  • Norman Levenberg
    • 1
    • 2
  1. 1.Department of Mathematics Swain Hall-EastIndiana UniversityBloomingtonUSA
  2. 2.Department of Mathematics and StatisticsUniversity of AucklandAucklandNew Zealand

Personalised recommendations