A Thought on Approximation by Bi-Analytic Functions

Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

A different approach to the problem of uniform approximations by the module of bi-analytic functions is outlined. This note follows the ideas from Khavinson (On a geometric approach to problems concerning Cauchy integrals and rational approximation. PhD thesis, Brown University, Providence, RI (1983), Proc Am Math Soc 101(3):475–483 (1987), Michigan Math J 34(3):465–473 (1987), Contributions to operator theory and its applications (Mesa, AZ, 1987). Birkhäuser, Basel (1988)), Gamelin and Khavinson (Am Math Mon 96(1):18–30 (1989)) and the more recent paper (Abanov et al. A free boundary problem associated with the isoperimetric inequality. arXiv:1601.03885, 2016 preprint), regarding approximation of \(\overline {z}\) by analytic functions.

2010 Mathematics Subject Classification.

30E10 30E99 

Notes

Acknowledgements

The author is indebted to the anonymous referee for several insightful remarks and references.

References

  1. 1.
    A. Abanov, C. Bénéteau, D. Khavinson, and R. Teodorescu. A free boundary problem associated with the isoperimetric inequality. arXiv:1601.03885, 2016. preprint.Google Scholar
  2. 2.
    Catherine Bénéteau and Dmitry Khavinson. The isoperimetric inequality via approximation theory and free boundary problems. Comput. Methods Funct. Theory, 6(2):253–274, 2006.Google Scholar
  3. 3.
    J. J. Carmona. Mergelyan’s approximation theorem for rational modules. Journal Approx. Theory, 44(2):113–126, 1985.Google Scholar
  4. 4.
    Peter L. Duren. Theory of H p Spaces, volume 38 of Pure and Applied Mathematics. Academic Press, New York-London, 1970.Google Scholar
  5. 5.
    M. Fleeman and D. Khavinson. Approximating \(\overline z\) in the Bergman space, to appear. volume 79 of Contemp. Math., pages 79–90. Amer. Math. Soc., Providence, RI. arXiv:1509.01370.Google Scholar
  6. 6.
    M. Fleeman and E. Lundberg. The Bergman analytic content of planar domains. arXiv:1602.03615.Google Scholar
  7. 7.
    T. W. Gamelin and D. Khavinson. The isoperimetric inequality and rational approximation. Amer. Math. Monthly, 96(1):18–30, 1989.Google Scholar
  8. 8.
    Zdeňka Guadarrama and Dmitry Khavinson. Approximating \(\overline z\) in Hardy and Bergman norms. In Banach spaces of analytic functions, volume 454 of Contemp. Math., pages 43–61. Amer. Math. Soc., Providence, RI, 2008.Google Scholar
  9. 9.
    Dmitry Khavinson. On uniform approximation by harmonic functions. Michigan Math. J., 34(3):465–473, 1987.Google Scholar
  10. 10.
    Dmitry Khavinson. Symmetry and uniform approximation by analytic functions. Proc. Amer. Math. Soc., 101(3):475–483, 1987.Google Scholar
  11. 11.
    Dmitry Khavinson. Duality and uniform approximation by solutions of elliptic equations. In Contributions to operator theory and its applications (Mesa, AZ, 1987), volume 35 of Oper. Theory Adv. Appl., pages 129–141. Birkhäuser, Basel, 1988.Google Scholar
  12. 12.
    Dmitry S. Khavinson. On a geometric approach to problems concerning Cauchy integrals and rational approximation. PhD thesis, Brown University, Providence, RI, 1983.Google Scholar
  13. 13.
    S. Ya. Khavinson. Foundations of the theory of extremal problems for bounded analytic functions and various generalizations of them. In Two Papers on Extremal Problems in Complex Analysis, volume 129 of Amer. Math. Soc. Transl. (2), pages 1–56. Amer. Math. Soc., Providence, RI, 1986.Google Scholar
  14. 14.
    M. Ya. Mazalov. Uniform approximations by bi-analytic functions on arbitrary compact sets in \(\mathbb C\). Mat. Sb., 195(5):79–102, 2004.Google Scholar
  15. 15.
    Anthony G. O’Farrell. Annihilators of rational modules. J. Functional Analysis, 19(4):373–389, 1975.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    P. V. Paramonov. Some new criteria for the uniform approximability of functions by rational fractions. Mat. Sb., 186(9):97–112, 1995.MathSciNetMATHGoogle Scholar
  17. 17.
    Tavan Trent and James Li Ming Wang. Uniform approximation by rational modules on nowhere dense sets. Proc. Amer. Math. Soc., 81(1):62–64, 1981.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tavan Trent and James Li Ming Wang. The uniform closure of rational modules. Bull. London Math. Soc., 13, 1981.Google Scholar
  19. 19.
    Joan Verdera. On the uniform approximation problem for the square of the Cauchy-Riemann operator. Pacific J. Math., 159(2):379–396, 1993.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    James Li Ming Wang. A localization operator for rational modules. Rocky Mountain J. Math., 19(4):999–1002, 1989.MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.University of South FloridaTampaUSA

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