Abstract:
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 $n$.
Keywords:Haar system of functions, Faber–Schauder system of functions, best
approximation of functions by polynomials, one-sided approximation
of functions by polynomials.