On uniform polynomial approximation of functions that are analytic in finite number of non-intersecting continuums

S.B. Vakarchuk (Academy of Customs Service of Ukraine), https://orcid.org/0000-0002-2562-8844
M.B. Vakarchuk (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0001-8606-9057
V.I. Zabutna (Oles Honchar Dnipropetrovsk National University)


We show that some of results, obtained by S.N. Bernstein, on constructive function theory, under certain conditions, take place for uniform polynomial approximation of functions that are analytic in finite number of non-intersecting continuums. On the base of obtained results for certain class of analytic functions we calculate asymptotic values of some $$$n$$$-widths.


Vakarchuk M.B. "On connection between complex spline-approximation and continuous differentiability of functions on curve", Izv. vuzov. Matematika, 1998; 432(5): pp. 3-5. (in Russian)

Vakarchuk S.B. "On widths of some classes of analytic functions. I", Ukrainian Math. J., 1992; 44(3): pp. 324-333. (in Russian) doi:10.1007/BF01063129

Vakarchuk S.B. "On the best mean approximation and super-convergence of sequence of polynomials of the best approximation", Ukrainian Math. J., 2000; 52(1): pp. 35-45. (in Russian) doi:10.1007/BF02514135

Vakarchuk S.B., Zabutna V.I. "Analytic splines and widths of functional classes", Proc. Int. Conf. "Approx. Theory and Comput. Math. Problems", dedic. to 75-year of state. univ., Dnipropetrovsk, 26-28.05.1993; p. 37. (in Russian)

Goluzin G.M. Geometric complex variable function theory, 1966. (in Russian)

Erokhin V.D. "On the best linear approximation of functions that are analytically extendable from given continuum to given domain", Uspekhi matem. nauk, 1968; 23(1): pp. 91-132. (in Russian) doi:10.1070/RM1968v023n01ABEH001234

Kiselev P.Ya. "On approximation of analytic functions by Faber-Walsh polynomials", Ukrainian Math. J., 1963; 15(2): pp. 193-199. (in Russian)

Kiselev P.Ya. "Some problems of approximation of analytic functions in finite number of domains", Researches on modern problems of constructive function theory, Baku, 1965; pp. 326-332. (in Russian)

Polya G., Szego G. Problems and theorems in analysis, Nauka, Moscow, 1978; vol. 1: 392 p. (in Russian)

Romaniuk V.S. "Kolmogorov widths of classes of functions that are analytic in bounded domains of the complex plane", Fourier series: theory and applications. Coll. scient. works, Kyiv, 1992; pp. 119-126. (in Russian)

Smirnov V.I., Lebedev N.A. Constructive theory of complex variable functions, Moscow-Leningrad, 1964; 438 p. (in Russian)

Suietin P.K. Series by Faber polynomials, 1984. (in Russian)

Tikhomirov V.M. Some problems of approximation theory, Izd-vo Moskovskogo un-ta, Moscow, 1976; 304 p. (in Russian)

Tikhomirov V.M. "Approximation Theory", Concl. of science and tech. VINITI. Modern problems of math. Fund. directions, 1987; 14: pp. 103-260. (in Russian)

Farkov Yu.A. "On Faber projector", Applic. of funct. analysis in approx. theory: Coll. scient. works, Kalinin, 1985; pp. 112-118. (in Russian)

Pinkus A. n-Widths in approximation theory, Springer-Verlag, Berlin, 1985; 290 p.

Walsh J.L. "Sur l'approximation par fonctions rationnelles et par fonctions holomorphes bornees", Ann. di Math. Pura et Appl., 1955; 39: pp. 267-277. (in French) doi:10.1007/BF02410774

Walsh J.L. "On the conformal mappings of multiply connected regions", Trans. Amer. Math. Soc., 1956; 82(1): pp. 128-146. doi:10.2307/1992982

Walsh J.L. "A generalization of Faber's polynomials", Matem. Ann., 1958; 136(1): pp. 23-33. doi:10.1007/BF01350283

DOI: https://doi.org/10.15421/240806



  • There are currently no refbacks.

Copyright (c) 2008 S.B. Vakarchuk, M.B. Vakarchuk, V.I. Zabutna

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991