On the best approximation in the metric of $L$ to certain classes of functions by Haar-system polynomials

Let $H_\omega$, $H_\omega^L$ be classes of functions $f(x)$ whose modulus of continuity $\omega(f;t)$ and, respectively, integral modulus of continuity $\omega(f;t)_L$ do not exceed a given modulus of continuity \omega(t)$, while$H_V$is a~class of functions$f(x)$whose variation$\mathop V\limits_0^1f$fdoes not exceed a~given number$V>0$. Bounds are obtained for the upper limit of the best approximations in the metric of$L$by Haar-system polynomials on the classes just introduced (on the class$H_\omega^L$only when$\omega(t)=Kt$). These bounds are exact for class$H_V$and, in case$\omega(t)$is convex, also for the classes$H_\omega$and$H\omega^L$.