New conditions for uniform approximation by polyanalytic polynomials
Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 227-241.

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We are interested in the problem of uniform approximability of functions by polyanalytic polynomials on compact subsets of the plane. We present new results showing the nature of the approximability conditions arising in this problem and their dependence on the order of polyanalyticity.
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J. J. Carmona; K. Yu. Fedorovskiy. New conditions for uniform approximation by polyanalytic polynomials. Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 227-241. http://geodesic.mathdoc.fr/item/TRSPY_2012_279_a14/

[1] Muskhelishvili N. I., Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, Noordhoff, Leyden, 1975 | MR | Zbl

[2] Balk M. B., Polyanalytic functions, Math. Res., 63, Akademie, Berlin, 1991 | MR | Zbl

[3] Mergelyan S. N., “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 ; Mergelyan S. N., Uniform approximations to functions of a complex variable, AMS Transl., 101, Amer. Math. Soc., Providence, RI, 1954 | MR | Zbl | MR

[4] Carmona J. J., “Mergelyan's approximation theorem for rational modules”, J. Approx. Theory, 44 (1985), 113–126 | DOI | MR | Zbl

[5] Fedorovskii K. Yu., “Uniform $n$-analytic polynomial approximations of functions on rectifiable contours in $\mathbb C$”, Math. Notes., 59 (1996), 435–439 | DOI | DOI | MR | Zbl

[6] Carmona J. J., Paramonov P. V., Fedorovskiy K. Yu., “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193 (2002), 1469–1492 | DOI | DOI | MR | Zbl

[7] Boivin A., Gauthier P. M., Paramonov P. V., “On uniform approximation by $n$-analytic functions on closed sets in $\mathbb C$”, Izv. Math., 68 (2004), 447–459 | DOI | DOI | MR | Zbl

[8] Carmona J. J., Fedorovskiy K. Yu., “Conformal maps and uniform approximation by polyanalytic functions”, Selected topics in complex analysis, Oper. Theory. Adv. Appl., 158, Birkhäuser, Basel, 2005, 109–130 | DOI | MR | Zbl

[9] Carmona J. J., Fedorovskiy K. Yu., “On the dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity”, Math. Notes., 83 (2008), 31–36 | DOI | DOI | MR | Zbl

[10] Davis P. J., The Schwarz function and its applications, Carus Math. Monogr., 17, Math. Assoc. Amer., Washington, D.C., 1974 | MR | Zbl

[11] Fedorovskii K. Yu., “On some properties and examples of Nevanlinna domains”, Proc. Steklov Inst. Math., 253, 2006, 186–194 | DOI | MR

[12] Baranov A. D., Fedorovskiy K. Yu., “Boundary regularity of Nevanlinna domains and univalent functions in model subspaces”, Sb. Math., 202 (2011), 1723–1740 | DOI | DOI | MR | Zbl

[13] Bishop E., “The structure of certain measures”, Duke Math. J., 25:2 (1958), 283–289 | DOI | MR | Zbl

[14] Stout E. L., The theory of uniform algebras, Bogden Quigley, Tarrytown-on-Hudson, NY, 1971 | MR | Zbl

[15] Paramonov P. V., “$C^m$-approximation by harmonic polynomials on compact sets in $\mathbb R^n$”, Russ. Acad. Sci., Sb. Math., 78 (1994), 231–251 | MR | Zbl

[16] Gustafsson B., Shapiro H. S., “What is a quadrature domain?”, Quadrature domains and their applications, Oper. Theory Adv. Appl., 156, Birkhäuser, Basel, 2005, 1–25 | DOI | MR | Zbl

[17] Aharonov D., Shapiro H. S., “Domains on which analytic functions satisfy quadrature identities”, J. anal. math., 30 (1976), 39–73 | DOI | MR | Zbl

[18] Sakai M., “Regularity of a boundary having a Schwarz function”, Acta math., 166 (1991), 263–297 | DOI | MR | Zbl

[19] Rudin W., Real and complex analysis, McGraw-Hill, New York, 1987 | MR | Zbl