Abstract
The concept of a Nevanlinna domain, which is the special analytic characteristic of a planar domain, has been naturally appeared in problems of uniform approximation by polyanalytic polynomials. In this paper we study this concept in connection with several allied approximation problems.
The author is partially supported by the Russian Foundation for Basic Research (grant no. 07-01-00503) and by the program “Leading Scientific Schools” (grant no. NSh-3877.2008.1).
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References
J.J. Carmona, K.Yu. Fedorovskiy and P.V. Paramonov, On uniform approximation by polyanalytic polynomials and Dirichlet problem for bianalytic functions, Sb. Math. 193(2002), no. 10, 1469–1492.
K.Yu. Fedorovski, Uniform n-analytic polynomial approximations of functions on rectifiable contours in C, Math. Notes., 1996, 59(4), 435–439.
K.Yu. Fedorovskiy, On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math., 2006, 253, 186–194.
T. Trent and J.L.-M. Wang, Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc., 1981, 81(1), 62–64.
J.J. Carmona, Mergelyan approximation theorem for rational modules, J. Approx. Theory 44 (1985), 113–126.
J.L. Wang, A localization operator for rational modules, Rocky Mountain J. of Math., 1989, 19(4), 999–1002.
K.Yu. Fedorovski, Approximation and boundary properties of polyanalytic functions, Proc. Steklov Inst. Math., 2001, 235, 251–260.
S.N. Mergelyan, Uniform approximation to functions of a complex variable, Amer. Math. Soc., Transl., 1954, 101; translation from Usp. Mat. Nauk, 1952, 7 (2), 31–122.
A. Boivin, P.M. Gauthier and P.V. Paramonov, Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications, Canadian J. of Math., 2002, 54(5), 945–969.
A. Boivin, P.M. Gauthier and P.V. Paramonov, Uniform approximation on closed subsets of ℂ by polyanalytic functions, Izv. Math., 2004, 68(3), 447–459.
A.B. Zajtsev, On uniform approximation of functions by polynomial solutions of second-order elliptic equations on plane compact sets, Izv. Math., 2004, 68(6), 1143–1156.
J.J. Carmona and K.Yu. Fedorovskiy, Conformal maps and uniform approximation by polyanalytic functions, Oper. Th. Adv. Appl., 2005, 158, 109–130.
J.J. Carmona and K.Yu. Fedorovskiy, On the dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity, Math. Notes, 2008, 83(1), 31–36.
H.S. Shapiro, Generalized analytic continuation, Symposia on Theor. Phys. and Math., 1968, 8, 151–163.
R.G. Douglas, H.S. Shapiro and A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Annales de l’Institut Fourier, 1970, 20(1), 37–76.
G.M. Goluzin, Geometric theory of functions of a complex variable, 2nd edition, “Nauka”, Moscow, 1966; English translation: Amer. Math. Soc., Providence, R.I., 1969.
Ch. Pommerenke, Boundary behaviours of conformal maps, Springer Verlag, Berlin, 1992.
A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer Verlag, Berlin, 2006.
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Fedorovskiy, K.Y. (2009). Nevanlinna Domains in Problems of Polyanalytic Polynomial Approximation. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_7
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DOI: https://doi.org/10.1007/978-3-7643-9906-1_7
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