Polynomial approximation in the mean on segments

Let $S_k$, $1\le k\le m$, be pairwise disjoint segments, $S_k = [a_k, b_k]$, $1\lt p_k\lt \infty$ functions $f_k$. Suppose that functions $f_k$ are defined on $S_k$, $f_k$ belongs to $C(S_k)$ and $f'_k$ belongs to $L^{P_k}(S_k)$. It is shown that for $n=1,2,\dots$ there are polynomials $P_n$, $ \deg P_n \le n$, that approximate all functions $f_k$ in the metric of $L^{P_k}$ with weights tending to infinity near the points $a_k$, $b_k$.