Hankel operators and problems of best approximation of unbounded functions

For each function $f$, $f\in VMO$, there exist a unique function $f_0$, analytic in the circle $\mathbb D$ and such that $\|f-f_0\|_\infty=\inf\{\|f-g\|_\infty\colon g\in VMO_A\}$. We define the operator of best approximation (nonlinear) $\mathcal A$, $\mathcal Af=f_0$, $f\in VMO$. In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes $X$, $X\subset VMO$, $\mathcal AX\subset X$. One investigates the classes $X$ containing unbounded functions. It is proved that if $P_-X$ is the space of the symbols of the Hankel operators from a Banach space $E$ of functions into the Hardy space $H^2$, then $\mathcal AX\subset X$. For $E$ one can take “almost” any space.