M.S. Saidusainov. Analysis of a theorem on the Jackson–Stechkin inequality in the Bergman space $B_2$ ... P. 217-224

Full text (in Russian)

We present a refinement of a theorem of V.A. Abilov, F.V. Abilova, and M.K. Kerimov on the exact constant in a Jackson type inequality between the mean-square approximation of a function of a complex variable by Fourier series in a system orthogonal in a bounded domain and the generalized modulus of continuity of order $m\geq 1$.

Keywords: generalized modulus of continuity, generalized translation operator, orthonormal system, Jackson-Stechkin inequality

Received Juny 28, 2018

Revised November 15, 2018

Accepted November 19, 2018

Mukim Saidusainovich Saidusaynov, Cand. Sci. (Phys.-Math.), University of Central Asia, Dushanbe, SPCE, 734013, Tajikistan, e-mail: smuqim@gmail.com


1.   Korneichuk N.P. The exact constant in D. Jackson’s theorem on best uniform approximation of continuous periodic functions. Sov. Math., Dokl., 1962, vol. 3, pp. 1040–1041.

2.   Chernykh N.I. On Jackson’s inequality in $L_2$. Proc. Steklov Inst. Math., 1967, vol. 88, pp. 75–78.

3.   Chernykh N.I. Best approximation of periodic functions by trigonometric polynomials in $L_2$. Math. Notes. 1967, vol. 2, no. 5, pp. 803–808. doi: 10.1007/BF01093942

4.   Zhuk V.V. Some exact inequalities between best approximations and moduli of continuity. Soviet Math. Dokl., 1971, vol. 12, pp. 223–226.

5.   Taikov L.V. Inequalities containing best approximations and the modulus of continuity of functions in $L_2$. Math. Notes, 1976, vol. 20, no. 3, pp. 797–800. doi: 10.1007/BF01097254

6.   Ligun A.A. Some inequalities between best approximation and moduli of continuity in $L_2$ space. Math. Notes, 1978, vol. 24, no. 6, pp. 917–921. doi: 10.1007/BF01140019

7.   Babenko A.G. The exact constant in the Jackson inequality in $L^2$. Math. Notes, 1986, vol. 39, no. 6, pp. 355–363. doi: 10.1007/BF01156673

8.   Ivanov V.I, Smirnov O.I. Konstanty Jeksona i konstanty Yunga v prostranstve $L_p$ [Jackson and Jung constants in the spaces $L_p$]. Tula: Tula State University Publ., 1995, 192 p.

9.   Shabozov M.S., Yusupov G.A. Best polynomial approximations in $L_2$ of classes of $2\pi$-periodic functions and exact values of their widths, Math. Notes, 2011, vol. 90, no. 5-6, pp. 748–757. doi: 10.1134/S0001434611110125

10.   Vakarchuk S.B., Zabutnaya V.I. Jackson — Stechkin type inequalities for special moduli of continuity and widths of function classes in the space $L_2$, Math. Notes, 2012, vol. 92, no. 3-4, pp. 458–472. doi: 10.1134/S0001434612090180

11.   Smirnov V.I., Lebedev N.A. Functions of a complex variable. Constructive theory. Cambridge, Mass.: M.I.T. Press, 1968, 488 p. ISBN: 9780262190466 . Original Russian text published in Smirnov V.I., Lebedev N.A. Konstruktivnaya teoriya funktsii kompleksnogo peremennogo. Moscow; Leningrad: Nauka Publ., 1964, 440 p.

12.   Abilov V.A., Abilova F.V., Kerimov M.K. Sharp estimates for the convergence rate of Fourier series of complex variable functions in $L_{2}(D,p(z))$. Comput. Mathematics and Mathematical Physics, 2010, vol. 50, no. 6, pp. 946–950. doi: 10.1134/S0965542510060023