Advertisement

Complex martingale convergence

  • G. A. Edgar
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1166)

Abstract

We investigate martingales appropriate for use in complex Banach spaces in connection with the complex uniform convexity popularized by Davis, Garling and Tomczak. This brings us into contact with diverse concepts, such as: pseudo-convex sets, plurisubharmonic functions, conformal martingales, the Radon-Nikodym property, and the analytic Randon-Nikodym property.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A.B. Aleksandrov, Essays on nonlocally convex Hardy classes. Lecture Notes in Math. 864, Springer-Verlag 1981, pp. 1–89.Google Scholar
  2. [2]
    J. Bourgain and W.J. Davis, Martingale transforms and complex uniform convexity. Preprint.Google Scholar
  3. [3]
    H.J. Bremermann, Complex convexity. Trans. Amer. Math. Soc. 82(1956) 17–51.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    _____, Holomorphic functionals and complex convexity in Banach spaces. Pacific J. Math. 7(1957)811–831.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A.V. Bukhvalov and A.A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach space. Mat. Zametki 31, No. 2 (1982) 203–214. English translation: Math. Notes 31 (1982) 104–110.MathSciNetzbMATHGoogle Scholar
  6. [6]
    W.J. Davis, D.J.H. Garling and N. Tomczak-Jaegermann, The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55(1984) 110–150.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Diestel and J.J. Uhl, Vector measures. Mathematical Surveys 15. American Mathematical Society 1977.Google Scholar
  8. [8]
    T. W. Gamelin, Uniform algebras and Jensen measures. Cambridge University Press, 1978.Google Scholar
  9. [9]
    D.J.H. Garling and N. Tomczak-Jaegermann, The cotype and uniform convexity of unitary ideals.Google Scholar
  10. [10]
    R.K. Getoor and M.J. Sharpe, Conformal martingales. Invent. Math. 16 (1972) 271–308.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. Globevnik, On complex strict and uniform convexity. Proc. Amer. Math. Soc. 47(1975)175–178.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. von Neumann, On rings of operators, reduction theory. Ann. of Math. (2) 50(1949)401–485.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Noverraz, Pseudo-convexité, Convexité Polynomiale et Domaines d'Holomorphie en Dimension Infinie. Mathematics Studies No. 3. North-Holland 1973.Google Scholar
  14. [14]
    J. Peetre, Locally analytically pseudo-convex topological vector spaces. Studia Math. 73(1982)253–262.MathSciNetzbMATHGoogle Scholar
  15. [15]
    T. Rado, Subharmonic functions. Springer-Verlag 1937.Google Scholar
  16. [16]
    L. Schwartz, Semi-Martingales sur des Variétés, et Martingales Conformes sur des Variétés Analytiques Complexes. Lecture Notes in Math 780. Springer-Verlag 1980.Google Scholar
  17. [17]
    E. Thorp and R. Whitley, The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Amer. Math. Soc. 18 (1967) 640–646.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • G. A. Edgar
    • 1
  1. 1.The Ohio State UniversityColumbus

Personalised recommendations