Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

On $ \mathcal{C}^m$-approximability of functions by polynomial solutions of elliptic equations on plane compact sets


Author: K. Yu. Fedorovskiĭ
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 677-689
MSC (2010): Primary 30E10, 41A30; Secondary 35J99
DOI: https://doi.org/10.1090/S1061-0022-2013-01260-X
Published electronically: May 24, 2013
MathSciNet review: 3088013
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Conditions of $ \mathcal {C}^m$-approximability of functions by polynomial solutions of homogeneous elliptic equations of order $ n$ on plane compact sets are studied. For positive integers $ m$ and $ n$ such that $ m\geq n-1$, new necessary and sufficient approximability conditions of a topological and metrical nature are obtained.


References [Enhancements On Off] (What's this?)

  • 1. M. B. Balk, Polyanalytic functions, Math. Res., vol. 63, Akademie-Verlag, Berlin, 1991. MR 1184141 (93k:30076)
  • 2. J. Verdera, $ \mathcal {C}^m$-approximation by solutions of elliptic equations, and Calderon-Zygmund operators, Duke Math. J. 55 (1987), no. 1, 157-187. MR 0883668 (88e:35011)
  • 3. -, Removability, capacity and approximation, Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419-473. MR 1332967 (96b:30086)
  • 4. P. V. Paramonov, Harmonic approximations in the $ C^1$-norm, Mat. Sb. 181 (1990), no. 10, 1341-1365; English transl., Math. USSR-Sb. 71 (1992), no. 1, 183-207. MR 1085885 (92j:31006)
  • 5. -, $ \mathcal {C}^m$-approximations by harmonic polynomials on compact sets in $ \mathbb{R}^n$, Mat. Sb. 184 (1993), no. 2, 105-128; English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 231-251. MR 1214947 (94d:42024)
  • 6. A. G. Bitsadze, Boundary value problems second order for elliptic equations, Nauka, Moscow, 1966; English transl., North-Holland Ser. in Appl. Math. Mech., vol. 5, North-Holland Publ. Co., Amsterdam, 1968. MR 0208152 (34:7962); MR 0226183 (37:1773)
  • 7. P. V. Paramonov, Approximations by harmonic polynomials in $ \mathcal {C}^1$-norm on compact sets in $ \mathbb{R}^2$, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 113-124; English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 321-331. MR 1230969 (94h:31001)
  • 8. P. V. Paramonov and K. Yu. Fedorovskiĭ, On uniform and $ \mathcal {C}^1$-approximability of functions on compact sets in $ \mathbb{R}^2$ by solutions of second-order elliptic equations, Mat. Sb. 190 (1999), no. 2, 123-144; English transl., Sb. Math. 190 (1999), no. 1-2, 285-307. MR 1701003 (2000k:41024)
  • 9. K. Yu. Fedorovskiy, $ \mathcal {C}^m$-approximation by polyanalytic polynomials on compact subsets of the complex plane, Complex Anal. Oper. Theory 5 (2011), no. 3, 671-681. MR 2836315 (2012j:30108)
  • 10. S. N. Mergelyan, Uniform approximations to functions of a complex variable, Uspekhi Mat. Nauk 7 (1952), no. 2, 31-122; English transl., Amer. Math. Soc. Transl., No. 101, Amer. Math. Soc., Providence, RI, 1954, 99 pp. MR 0051921 (14:547e)
  • 11. J. J. Carmona, P. V. Paramonov, and K. Yu. Fedorovskiĭ, Uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Mat. Sb. 193 (2002), no. 10, 75-98; English transl., Sb. Math. 193 (2002), no. 9-10, 1469-1492. MR 1937036 (2004f:30027)
  • 12. A. Boivin, P. M. Gauthier, and P.V. Paramonov, On uniform approximation by $ n$-analytic functions on closed sets in $ \mathbb{C}$, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 3, 15-28; English transl., Izv. Math. 68 (2004), no. 3, 447-459. MR 2069192 (2005m:30044)
  • 13. J. J. Carmona and K. Yu. Fedorovskiy, Conformal maps and uniform approximation by polyanalytic functions, Selected Topics in Complex Analysis, Oper. Theory: Adv. Appl., vol. 158, Birkhäuser Verlag, Basel, 2005, pp. 109-130. MR 2147592 (2006b:30067)
  • 14. -, On the dependence of conditions for the uniform approximability of functions by polyanalytic polynomials on the order of polyanalyticity, Mat. Zametki 83 (2008), no. 1, 32-38; English transl., Math. Notes 83 (2008), no. 1-2, 31-36. MR 2399995 (2009c:30012)
  • 15. A. B. Zaĭtsev, On the uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact sets in $ \mathbb{R}^2$, Mat. Zametki 74 (2003), no. 1, 41-51; English transl., Math. Notes 74 (2003), no. 1-2, 38-48. MR 2010675 (2005a:41015)
  • 16. -, On the uniform approximability of functions by polynomial solutions of second-order elliptic equations on planar compact sets, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 6, 85-98; English transl., Izv. Math. 68 (2004), no. 6, 1143-1156. MR 2108523 (2005h:41050)
  • 17. -, Uniform approximation of second-order elliptic equations by polynomial solutions and the corresponding Dirichlet problem, Tr. Mat. Inst. Steklova 253 (2006), 67-80; English transl., Proc. Steklov Inst. Math. 2006, no. 2 (253), 57-70. MR 2338688 (2008f:35009)
  • 18. J. F. Treves, Lectures on linear partial differential equations with constant coefficients, Notas de Matemátiea, No. 27, Inst. Mat. Pura Apl., Rio de Janeiro, 1961. MR 0155078 (27:5020)
  • 19. S. K. Smirnov and V. P. Khavin, Approximation and extension problems for some classes of vector fields, Algebra i Analiz 10 (1998), no. 3, 133-162; English transl., St. Petersburg Math. J. 10 (1999), no. 3, 507-528. MR 1628034 (99f:41046)
  • 20. L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren Math. Wiss., Bd. 256, Springer-Verlag, Berlin, 1983. MR 0717035 (85g:35002a)
  • 21. N. N. Tarkhanov, Laurent series for solutions of elliptic systems, Nauka, Novosibirsk, 1991. (Russian) MR 1226897 (94e:35013)
  • 22. A. G. Vitushkin, Analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141-199; English transl., Russian Math. Surveys 22 (1967), 139-200. MR 0229838 (37:5404)
  • 23. P. V. Paramonov and J. Verdera, Approximation by solutions of elliptic equations on closed subsets of euclidean space, Math. Scand. 74 (1994), 249-259. MR 1298365 (95i:41040)
  • 24. M. Ya. Mazalov, A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations, Mat. Sb. 199 (2008), no. 1, 15-46; English transl., Sb. Math. 199 (2008), no. 1-2, 13-44. MR 2410145 (2009c:35005)
  • 25. R. Narasimkhan, Analysis on real and complex manifolds, 2nd ed., Adv. Stud. Pure Math., vol. 1, North-Holland Publ. Co., Amsterdam-London, 1973. MR 0346855 (49:11576)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 30E10, 41A30, 35J99

Retrieve articles in all journals with MSC (2010): 30E10, 41A30, 35J99


Additional Information

K. Yu. Fedorovskiĭ
Affiliation: Bauman Moscow State Technical University, Moscow 105005, Russia
Email: kfedorovs@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01260-X
Keywords: Homogeneous elliptic operator, $L$-analytic function, $L$-analytic polynomial, $\mathcal{C}^m$-approximation, localization operator
Received by editor(s): January 25, 2012
Published electronically: May 24, 2013
Additional Notes: The author was partially supported by RFBR (grant nos. 12-01-00434-a and 10-01-00837-a)
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society