Abstract
Conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets of special form in ℝ2 are studied. The results obtained are of analytic character. Conditions of solvability and uniqueness for the corresponding Dirichlet problem are also studied. It is proved that the polynomial approximability on the boundary of a domain is not generally equivalent to the solvability of the corresponding Dirichlet problem.
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References
P. V. Paramonov and K. Yu. Fedorovskii, “Uniform and C 1-Approximability of Functions on Compact Subsets of ℝ2 by Solutions of Second-Order Elliptic Equations,” Mat. Sb. 190(2), 123–144 (1999) [Sb. Math. 190, 285–307 (1999)].
P. V. Paramonov, “C m-Approximations by Harmonic Polynomials on Compact Sets in ℝn,” Mat. Sb. 184(2), 105–128 (1993) [Sb. Math. 78 (1), 231–251 (1994)].
A. B. Zaitsev, “Uniform Approximability of Functions by Polynomials of Special Classes on Compact Sets in ℝ2,” Mat. Zametki 71(1), 75–87 (2002) [Math. Notes 71, 68–79 (2002)].
J. J. Carmona, P. V. Paramonov, and K. Yu. Fedorovskii, “On Uniform Approximation by Polyanalytic Polynomials and the Dirichlet Problem for Bianalytic Functions,” Mat. Sb. 193(10), 75–98 (2002) [Sb. Math. 193, 1469–1492 (2002)].
A. Boivin, P. M. Gauthier, and P. V. Paramonov, “On Uniform Approximation by n-Analytic Functions on Closed Sets in ℂ,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(3), 15–28 (2004) [Izv. Math. 68, 447–459 (2004)].
A. B. Zaitsev, “Uniform Approximability of Functions by Polynomial Solutions of Second-Order Elliptic Equations on Compact Plane Sets,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(6), 85–98 (2004) [Izv. Math. 68, 1143–1156 (2004)].
K. Yu. Fedorovskii, “Uniform n-Analytic Polynomial Approximations of Functions on Rectifiable Contours in ℂ,” Mat. Zametki 59(4), 604–610 (1996) [Math. Notes 59, 435–439 (1996)].
K. Yu. Fedorovskii, “On Some Properties and Examples of Nevanlinna Domains,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 253, 204–213 (2006) [Proc. Steklov Inst. Math. 253, 186–194 (2006)].
P. Davis, The Schwarz Function and Its Applications (Math. Assoc. Am., Washington, DC, 1974), Carus Math. Monogr. 17.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2: Functional Analysis (Academic, New York, 1972; Mir, Moscow, 1977).
I. I. Privalov, Boundary Properties of Analytic Functions (Gostekhizdat, Moscow, 1950) [in Russian].
P. V. Paramonov and J. Verdera, “Approximation by Solutions of Elliptic Equations on Closed Subsets of Euclidean Space,” Math. Scand. 74(2), 249–259 (1994).
G. C. Verchota and A. L. Vogel, “Nonsymmetric Systems on Nonsmooth Planar Domains,” Trans. Am. Math. Soc. 349(11), 4501–4535 (1997).
M. B. Balk, Polyanalytic Functions (Akademie, Berlin, 1991), Math. Res. 63.
A. B. Zaitsev, “Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in ℝ2,” Mat. Zametki 74(1), 41–51 (2003) [Math. Notes 74, 38–48 (2003)].
T. W. Gamelin, Uniform Algebras (Prentice Hall, Englewood Cliffs, NJ, 1969; Mir, Moscow, 1973).
L. V. Ahlfors, Lectures on Quasiconformal Mappings (Van Nostrand, Princeton, NJ, 1966; Mir, Moscow, 1969).
J. J. Carmona, “Mergelyan Approximation Theorem for Rational Modules,” J. Approx. Theory 44, 113–126 (1985).
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Original Russian Text © A.B. Zaitsev, 2006, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 253, pp. 67–80.
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Zaitsev, A.B. Uniform approximation by polynomial solutions of second-order elliptic equations, and the corresponding Dirichlet problem. Proc. Steklov Inst. Math. 253, 57–70 (2006). https://doi.org/10.1134/S0081543806020064
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DOI: https://doi.org/10.1134/S0081543806020064