Approximation of solutions of the equation $\overline\partial^jf=0$, $j\geqslant1$, in domain with quasiconformal boundary

This article establishes direct and inverse theorems of approximation theory (of the same type as theorems of Dzyadyk) that describe the quantitative connection between the smoothness properties of solutions of the equation
$$\overline\partial^jf=0,\qquad j\geqslant1,$$
and the rate of their approximation by “module” polynomials of the form
$$ P_N(z)=\sum_{n=0}^{j-1}\sum_{m=0}^{N-n}a_{m,n}z^m\overline z^n,\qquad N\geqslant j-1. $$
In particular, a constructive characterization is obtained for generalized Hölder classes of such functions on domains with quasiconformal boundary.
Bibliography: 19 titles.