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Kolmogorov's type inequalities in Bergman space $B_2$ and some of its applications

D. K. Tukhliev

Khujand State University (Khujand, Tajikistan)

Abstract: Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{Z_{+}}$ be the set of non-negative integers, $\mathbb{C}$ be the set of complex numbers, $A(U)$ be the set of analytic functions in the unit circle
$U:=\left\{z\in \mathbb{C}:|z|<1\right\}$, $B_2$ – be the Bergman spaces of functions $f\in A(U)$, endowed with a finite norm
$$\|f\|_2:=\|f\|_{B_2}=\left(\frac{1}{\pi}\displaystyle\iint_{(U)}|f(z)|^2d\sigma\right)^{1/2}.$$
For $f\in A(U)$, we denote the usual derivative of order $m\in \mathbb{N}$ by $f^{(m)}(z)$ and introduce a class of functions
$$B^{(m)}_2:=\left\{f\in B_2:\|f^{(m)}\|_2<\infty\right\}.$$
Let $E_{n-1}(f)_2$ be the magnitude of the best approximation of function $f\in B_2$ by complex algebraic polynomials of degree $\leq n-1.$ In this paper, a number of exact inequalities are found between the value of the best approximation of intermediate derivatives $E_{n-\nu-1}(f^{(\nu)})_2$ $(\nu=1,2,\cdots,m-1; m\geq2)$ and the best approximation $E_{n-m-1}(f^{(m)})_2$ of the highest derivative $f^{(m)}.$ Let $W^{(m)}_2:=W^{(m)}_2(U) \hspace{1mm} (m\in \mathbb{N})$ be a class of functions $f\in B^{(m)}_2$ for which $\|f^{(m)}\|_2\leq 1$. In this paper is proved that for any $n,m\in \mathbb{N}, \nu\in\mathbb{Z_+}, n>m\geq\nu$, the equality of takes place
$$E_{n-\nu-1}(W^{(m)}_2)_2=\sup\{E_{n-\nu-1}(f^{(\nu)})_2: f\in W^{(m)}_2\}= \frac{\alpha_{n,\nu}}{\alpha_{n,m}}\cdot\sqrt{\frac{n-m+1}{n-\nu+1}},$$
and also, in the space $B_2$ for functions $f\in B^{(m)}_2$ for all $1\leq\nu\leq m-1, m\geq2$, an exact inequality of the Kolmogorov type
$$ E_{n-\nu-1}(f^{(\nu)})_2\leq A_{m,\nu}(n)(E_{n-1}(f)_2)^{1-\nu/m}\cdot(E_{n-m-1}(f^{(m)})_2)^{\nu/m},$$
is found, where the constant $A_{m,\nu}(n)$ is explicitly written out. Some applications of the resulting inequality are given.

Keywords: Bergman space, exact inequalities, mean-square approximations, best polynomial approximation, extremal problems, Kolmogorov type inequality.

UDC: 517.5

Received: 27.07.2023
Accepted: 21.12.2023

DOI: 10.22405/2226-8383-2023-24-5-228-236