On best approximation of functions in Bergman space $B_{2}$
D. K. Tukhliev Khujand State University named after academician Bobojon Gafurov, Mavlonbekova str. 1, 735700, Khujand, Tajikistan
Abstract:
In the paper we study extremal problems related to the best polynomial approximation of functions analytic in the unit disk and belonging to the Hilbert Bergman space
$B_2$. We find exact inequalities for the best approximation of an arbitrary function
$f\in B_2$, analytic in the unit disk, by algebraic complex polynomials
$p_n\in \mathcal{P}_n$ by means of the averaged value of the modulus of continuity
$\omega(f^{(r)},t)_{B_2}$ of the
$r$th derivative
$f^{(r)}$ in the space
$B_2$. We introduce the class
$W^{(r)}_2(\omega,\Phi)$ of functions analytic in the unit disk whose averaged value of the modulus of continuity of the derivative
$f^{(r)}$ satisfies the inequality
$$ \int\limits_{0}^{u}\omega^2(f^{(r)},t)_{B_2}\sin\frac{\pi}{u}t\,d t\leq \Phi^2(u), 0\leq u\leq 2\pi. $$
For certain restrictions for majorant
$\Phi$, we calculate exact values of various
$n$–widths for the introduced class of functions. To solve the mentioned problems, we use the methods of solving extremal problems in normed spaces and we use the method for estimating
$n$–widths developed by V.M. Tikhomirov.
Keywords:
extremal problems, approximation of functions, modulus of continuity, suprema, $n$–widths, Bergman space.
UDC:
517.5
MSC: 41A17,
41A25 Received: 30.09.2024
Fulltext:
PDF file (448 kB)
