Abstract
Extreme problems related to the best simultaneous polynomial approximation of analytic functions in the unit disc belonging to the Bergman space \({{B}_{2}}\) are studied. Here, a number of exact theorems and the exact values of the upper bounds of the best simultaneous approximations of functions and their consecutive derivatives by polynomials and their corresponding derivatives on some classes of complex functions belonging to the Bergman space \({{B}_{2}}\) are obtained.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
M. Z. Dveirin and I. V. Chebanenko, “On polynomial approximation in Banach spaces of analytical functions,” in Theory of Mappings and Approximation of Functions (Naukova Dumka, Kiev, 1983).
C. Horowitz, “Zeros of functions in Bergman Space,” Bull. Am. Math. Soc. 80, 713–714 (1974). https://doi.org/10.1090/S0002-9904-1974-13563-9
S. B. Vakarchuk, “Diameters of certain classes of functions analytic in the unit disc. I,” Ukr. Math. J. 42, 769–778 (1990). https://doi.org/10.1007/BF01062078
S. B. Vakarchuk, “On diameters of certain classes of functions analytic in the unit disc. II,” Ukr. Math. J. 42, 907–914 (1990). https://doi.org/10.1007/BF01099219
S. B. Vakarchuk, “Best linear methods of approximation and widths of classes of analytic functions in a disk,” Math. Notes 57, 21–27 (1995). https://doi.org/10.1007/BF02309390
S. B. Vakarchuk, “On the best linear approximation methods and the widths of certain classes of analytic functions,” Math. Notes 65, 153–158 (1999). https://doi.org/10.1007/BF02679811
S. B. Vakarchuk, “On inequalities of Kolmogorov type for some Banach spaces of analytic functions,” in Some Questions of Analysis and Differential Topology (Naukova Dumka, Kiev, 1988).
M. Sh. Shabozov and M. S. Saidusainov, “Upper bounds for the approximation of certain classes of functions of a complex variable by Fourier series in the space L 2 and n-widths,” Math. Notes 103, 656–668 (2018). https://doi.org/10.1134/S0001434618030343
M. Sh. Shabozov and M. S. Saidusainov, “Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 25, 258–272 (2019). https://doi.org/10.21538/0134-4889-2019-25-2-258-272
Kh. M. Khuromonov and M. Sh. Shabozov, “Jackson–Stechkin type inequalities between the best joint polynomials approximation and a smoothness characteristic in Bergman space,” Vladikavkazsk. Mat. Zh. 24 (1), 109–120 (2022). https://doi.org/10.46698/d2512-2100-1282-i
V. N. Malozemov, Combined Approximation of Functions and Its Derivatives (Izd-vo Leningrad. Gos. Univ., Leningrad, 1973).
S. B. Vakarchuk and V. I. Zabutnaya, “Jackson–Stechkin type inequalities for special moduli of continuity and widths of function classes in the space L 2,” Math. Notes 92, 458–472 (2012). https://doi.org/10.1134/S0001434612090180
M. Sh. Shabozov, G. A. Yusupov, and J. J. Zargarov, “On the best simultaneous polynomial approximation of functions and their derivatives in Hardy spaces,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 27, 239–254 (2021). https://doi.org/10.21538/0134-4889-2021-27-4-239-254
V. I. Smirnov and N. A. Lebedev, Constructive Theory of Functions of Complex Variable (Nauka, Moscow, 1964).
L. V. Taikov, “Inequalities containing best approximations and the modulus of continuity of functions in L 2,” Math. Notes Acad. Sci. USSR 20, 797–800 (1976). https://doi.org/10.1007/BF01097254
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
About this article
Cite this article
Khuromonov, K.M. On the Best Simultaneous Approximation of Functions in the Bergman Space B2. Russ Math. 67, 50–59 (2023). https://doi.org/10.3103/S1066369X23050055
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X23050055