Full text (in Russian)
Suppose that $\textbf{z}:=(\xi,\zeta)=(re^{it},\rho e^{i\tau})$, where $0\leq r,\rho<\infty$ and $0\leq t,\tau\leq 2\pi$, is a point in the two-dimensional complex space $\mathbb{C}^{2}$; $U^{2}:=\{\textbf{z}\in\mathbb{C}^{2}: |\xi|<1,\,|\zeta|<1\}$ is the unit bidisk in $\mathbb{C}^{2}$; $\mathcal{A}(U^{2})$ is the class of functions analytic in $U^{2}$; and $B_{2}:=B_{2}(U^{2})$ is the Bergman space of functions $f\in\mathcal{A}(U^{2})$ such that
$$
\|f\|_{2}:=\|f\|_{B_{2}(U^{2})}=\left(\frac{1}{4\pi^{2}}\iint_{(U^{2})}|f(\xi,\zeta)|^{2}d\sigma_{\xi}d\sigma_{\zeta}\right)^{1/2}<+\infty,
$$
where $d\sigma_{\xi}:=dxdy$, $d\sigma_{\zeta}:=dudv$, and the integral is understood in the Lebesgue sense. S.B. Vakarchuk and M.B. Vakarchuk (2013) proved that, under some conditions on the Taylor coefficients $c_{pq}(f)$ in the expansion of $f(\xi,\zeta)$ in a double Taylor series, the following exact Kolmogorov inequality holds:
$$
\left\|f^{(k-\mu,l-\nu)}\right\|_{2}\leq \mathcal{C}_{k,l}(\mu,\nu) \,\|f\|_{2}^{\mu\nu/(kl)}\,\left\|f^{(k,0)}\right\|_{2}^{(1-\mu/k)\nu/l}\,\left\|f^{(0,l)}\right\|_{2}^{(1-\nu/l)\mu/k}\,\left\|f^{(k,l)}\right\|_{2}^{(1-\mu/k)(1-\nu/l)},
$$
where the numerical coefficients $\mathcal{C}_{k,l}(\mu,\nu)$ are explicitly defined by the parameters $k,l\in\mathbb{N}$ and $\mu,\nu\in\mathbb{Z}_{+}$. We find an exact Kolmogorov type inequality for the best approximations $\mathscr{E}_{m-1,n-1}(f)_{2}$ of functions $f\in B_{2}(U^{2})$ by generalized polynomials (quasipolynomials):
$$
\mathscr{E}_{m-k+\mu-1,n-l+\nu-1}\big(f^{(k-\mu,l-\nu)}\big)_{2}
$$
$$
{}\leq\frac{\alpha_{m,k-\mu}\alpha_{n,l-\nu}(m-k+1)^{(k-\mu)/(2k)}(n-l+1)^{(l-\nu)/(2l)}(m+1)^{\mu/(2k)}(n+1)^{\nu/(2l)}}{(\alpha_{m,k})^{1-\mu/m}(\alpha_{n,l})^{1-\nu/l}\left[(m-k+\mu+1)(n-l+\nu+1)\right]^{1/2}} $$
$$
{}\times\big(\mathscr{E}_{m-1,n-1}(f)_{2}\big)^{\frac{\mu\nu}{kl}}\big(\mathscr{E}_{m-k-1,n-l}\big(f^{(k,0)}\big)_{2}\big)^{(1-\frac{\mu}{k})\frac{\nu}{l}}
$$
$$
{}\times\big(\mathscr{E}_{m-1,n-l-1}\big(f^{(0,l)}\big)_{2}\big)^{\frac{\mu}{k}(1-\frac{\nu}{l})}\big(\mathscr{E}_{m-k-1,n-l-1}\big(f^{(k,l)}\big)_{2}\big)^{(1-\frac{\mu}{k})(1-\frac{\nu}{l})}
$$
in the sense that there exists a function $f_{0}\in B_{2}^{(k,l)}$ for which the inequality turns into an equality.
Keywords: Kolmogorov type inequality, Bergman space, analytic function, quasipolynom, upper bound
Received July 3, 2018
Revised October 19, 2018
Accepted October 22, 2018
Mirgand Shabozovich Shabozov. Dr. Phys.-Math. Sci., Prof., Tajik National University, Dushanbe, 734025 Republic of Tajikistan, e-mail: shabozov@mail.ru
Vosif Dodkhudoevich Sainakov. Technological University of Tajikistan, Dushanbe, 734061 Republic of Tajikistan, e-mail: vosifvoiz@mail.ru
REFERENCES
1. Babenko V.F., Korneichuk N.P., Kofanov V.A. and Pichugov S.A. Neravenstva dlya proizvodnykh i ikh prilozheniya [Inequalities for derivatives and their applications]. Kiev: Naukova dumka, 2003, 590 p. ISBN: 966-00-0074-4 .
2. Arestov V.V. Approximation of unbounded operators by bounded operators and related extremal problems. Russ. Math. Surv., 1996, vol. 51, no. 6, pp. 1093–1126. doi: 10.1070/RM1996v051n06ABEH003001
3. Vakarchuk S.B. On inequalities of Kolmogorov type for some Banach spaces of analytic functions. Nekotorye voprosy analiza i differentsial’noi topologii [Some questions of analysis and differential topology], Collect. Sci. Works, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1988, pp. 4–7 (in Russian).
4. Shabozov M.Sh., Saidusainov M.S. Inequality of Kolmogorov type in the weighted Bergman space. Reports of the Academy of Sciences of the Republic of Tajikistan, 2007, vol. 50, no. 1, pp. 14–19 (in Russian).
5. Vakarchuk S.B., Vakarchuk M.B. On inequalities of Kolmogorov type for analytic functions in a disk. Dnipr. Univ. Math. Bull., 2012, vol. 17, no. 6/1, pp. 82–88 (in Russian).
6. Saidusainov M.S. Exact inequalities of Kolmogorov type for functions belonging to the weighted Bergman space, Proc. Internat. Summer Math. Stechkin School-Conf. on Function Theory, Tajikistan, Dushanbe, 15-25 August, 2016, pp. 217–223 (in Russian). ISBN: 978-9-9975-9175-3.
7. Vakarchuk S.B., Vakarchuk M.B. Inequalities of Kolmogorov type for analytic functions of one and two complex variables and their application to approximation theory. Ukr. Math. J., 2012, vol. 63, no. 12, pp. 1795–1819. doi: 10.1007/s11253-012-0615-3
8. Vakarchuk S.B., Vakarchuk M.B. On inequalities of Kolmogorov type for analytic functions in the unit bicircle. Dnipr. Univ. Math. Bull., 2013, vol. 18, no. 6/1, pp. 61–66 (in Russian).
9. Brudnyi Yu.A. Approximation of functions of n variables by quasipolynomials. Math. USSR-Izv., 1970, vol. 4, no. 3, pp. 568–586. doi: 10.1070/IM1970v004n03ABEH000922
10. Potapov M.K. On approximation by “angle”. Proc. Conf. Constructive Theory of Functions, Budapest, 1972, pp. 371–399 (in Russian).
11. Shabozov M.Sh., Vakarchuk S.B. On exact values of quasiwidths of some classes of functions. Ukr. Math. J., 1996, vol. 48, no. 3, pp. 338–346. doi: 10.1007/BF02378524
12. Shabozov M.Sh., Akobirshoev M. Quasiwidths of some classes of differentiable periodic functions of two variables. Dokl. Akad. Nauk, 2005, vol. 404, no. 4, pp. 460–464 (in Russian).
13. Smirnov V.I., Lebedev N.A. Functions of a complex variable. Constructive theory. London: Iliffe Books Ltd., 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.
14. Hardy G.H., Littlewood J.E., P$\acute{\mathrm{o}}$lya G. Inequalities. Cambridge: Cambridge University Press, 1934, 340 p. ISBN(2nd ed.): 0-521-05206-8 . Translated to Russian under the title Neravenstva. Moscow: Inostr. Lit. Publ., 1948, 456 p.
15. Shabozov M.Sh., Saidusaynov M.S. 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, 2018, vol. 103, no. 3-4, pp. 656–668. doi: 10.1134/S0001434618030343
