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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 4, Pages 270–282 (Mi timm1592)

On Kolmogorov type inequalities in the Bergman space for functions of two variables

M. Sh. Shabozova, V. D. Sainakovb

a Tajik National University, Dushanbe
b Tajik Technological University

Abstract: Suppose that $\mathrm{z}:=(\xi,\zeta)=(re^{it},\rho e^{i\tau})$, where $0\leq r,\rho<\infty$ and $0\leq t,\tau\leq 2\pi$, is a point in the two-dimensional complex space $\mathbb{C}^{2}$; $U^{2}:=\{\mathrm{z}\in\mathbb{C}^{2}: |\xi|<1, |\zeta|<1\}$ 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
$$\|f\|_{2}:=\|f\|_{B_{2}(U^{2})}=(\frac{1}{4\pi^{2}}\iint_{(U^{2})}|f(\xi,\zeta)|^{2}d\sigma_{\xi}d\sigma_{\zeta})^{1/2}<+\infty,$$
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:
$$\|f^{(k-\mu,l-\nu)}\|_{2}\leq \mathcal{C}_{k,l}(\mu,\nu) \|f\|_{2}^{\mu\nu/(kl)} \|f^{(k,0)}\|_{2}^{(1-\mu/k)\nu/l} \|f^{(0,l)}\|_{2}^{(1-\nu/l)\mu/k} \|f^{(k,l)}\|_{2}^{(1-\mu/k)(1-\nu/l)},$$
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):
$$\mathscr{E}_{m-k+\mu-1,n-l+\nu-1}(f^{(k-\mu,l-\nu)})_{2}$$

$$\leq\frac{\alpha_{m,k-\mu}\alpha_{n,l-\nu}(m-k+1)^{(k-\mu)/(2k)}(n-l+1)^{(l-\nu)/(2l)}(m+1)^{\mu/(2k)}(n+1)^{\nu/(2l)}}{(\alpha_{m,k})^{1-\mu/m}(\alpha_{n,l})^{1-\nu/l}[(m-k+\mu+1)(n-l+\nu+1)]^{1/2}}$$

$$\times(\mathscr{E}_{m-1,n-1}(f)_{2})^{\frac{\mu\nu}{kl}}(\mathscr{E}_{m-k-1,n-l}(f^{(k,0)})_{2})^{(1-\frac{\mu}{k})\frac{\nu}{l}}$$

$$\times(\mathscr{E}_{m-1,n-l-1}(f^{(0,l)})_{2})^{\frac{\mu}{k}(1-\frac{\nu}{l})}(\mathscr{E}_{m-k-1,n-l-1}(f^{(k,l)})_{2})^{(1-\frac{\mu}{k})(1-\frac{\nu}{l})}$$
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.

DOI: https://doi.org/10.21538/0134-4889-2018-24-4-270-282

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Bibliographic databases:

UDC: 517.5
MSC: 42C10, 47A58, 30E10, 32E05
Revised: 19.10.2018
Accepted:22.10.2018

Citation: M. Sh. Shabozov, V. D. Sainakov, “On Kolmogorov type inequalities in the Bergman space for functions of two variables”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 270–282

Citation in format AMSBIB
\Bibitem{ShaSai18} \by M.~Sh.~Shabozov, V.~D.~Sainakov \paper On Kolmogorov type inequalities in the Bergman space for functions of two variables \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 4 \pages 270--282 \mathnet{http://mi.mathnet.ru/timm1592} \crossref{https://doi.org/10.21538/0134-4889-2018-24-4-270-282} \elib{https://elibrary.ru/item.asp?id=36517716} 

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This publication is cited in the following articles:
1. M. Sh. Shabozov, M. O. Akobirshoev, “O neravenstvakh tipa Kolmogorova dlya periodicheskikh funktsii dvukh peremennykh v $L_2$”, Chebyshevskii sb., 20:2 (2019), 348–365
2. O. A. Dzhurakhonov, “Priblizhenie funktsii dvukh peremennykh «krugovymi» summami Fure — Chebysheva v $L_{2,\rho}$”, Vladikavk. matem. zhurn., 22:2 (2020), 5–17
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