Estimates for the error of approximation of functions in $L_p^1$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems

We find exact estimates for the error of approximation of functions in the classes $L_p^1$ by polynomials in the Haar system and partial sums of the Faber–Schauder series in the metrics of the spaces $L_p$. The error in approximating a function $f\in L_p^1$ is estimated in terms of the norms of the first derivatives $\|f^{(1)}\|_{L_p}$ and $\|f^{(1)}-\overline S^{(1)}_n(f)\|_{L_p}$. The resulting bounds are unimprovable for some values of $n$.