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.
Dedicated to the memory of André Boivin, a kind and gentle friend.
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References
A. Abanov, C. Bénéteau, D. Khavinson, and R. Teodorescu. A free boundary problem associated with the isoperimetric inequality. arXiv:1601.03885, 2016. preprint.
Catherine Bénéteau and Dmitry Khavinson. The isoperimetric inequality via approximation theory and free boundary problems. Comput. Methods Funct. Theory, 6(2):253–274, 2006.
J. J. Carmona. Mergelyan’s approximation theorem for rational modules. Journal Approx. Theory, 44(2):113–126, 1985.
Peter L. Duren. Theory of H p Spaces, volume 38 of Pure and Applied Mathematics. Academic Press, New York-London, 1970.
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.
M. Fleeman and E. Lundberg. The Bergman analytic content of planar domains. arXiv:1602.03615.
T. W. Gamelin and D. Khavinson. The isoperimetric inequality and rational approximation. Amer. Math. Monthly, 96(1):18–30, 1989.
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.
Dmitry Khavinson. On uniform approximation by harmonic functions. Michigan Math. J., 34(3):465–473, 1987.
Dmitry Khavinson. Symmetry and uniform approximation by analytic functions. Proc. Amer. Math. Soc., 101(3):475–483, 1987.
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.
Dmitry S. Khavinson. On a geometric approach to problems concerning Cauchy integrals and rational approximation. PhD thesis, Brown University, Providence, RI, 1983.
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.
M. Ya. Mazalov. Uniform approximations by bi-analytic functions on arbitrary compact sets in \(\mathbb C\). Mat. Sb., 195(5):79–102, 2004.
Anthony G. O’Farrell. Annihilators of rational modules. J. Functional Analysis, 19(4):373–389, 1975.
P. V. Paramonov. Some new criteria for the uniform approximability of functions by rational fractions. Mat. Sb., 186(9):97–112, 1995.
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.
Tavan Trent and James Li Ming Wang. The uniform closure of rational modules. Bull. London Math. Soc., 13, 1981.
Joan Verdera. On the uniform approximation problem for the square of the Cauchy-Riemann operator. Pacific J. Math., 159(2):379–396, 1993.
James Li Ming Wang. A localization operator for rational modules. Rocky Mountain J. Math., 19(4):999–1002, 1989.
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The author is indebted to the anonymous referee for several insightful remarks and references.
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Khavinson, D. (2018). A Thought on Approximation by Bi-Analytic Functions. In: Mashreghi, J., Manolaki, M., Gauthier, P. (eds) New Trends in Approximation Theory. Fields Institute Communications, vol 81. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7543-3_6
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