Polynomial approximation in the mean on segments

Let $S_k$, $1\le k\le m$ – pairs of disjoint segments, $S_k = [a_k, b_k]$, $1<p_k<\infty$ functions $f_k$ are defined on $S_k$, $f_k$ belongs to $C(S_k)$ and $f'_k$ belongs to $L^{P_k}(S_k)$. The work proves 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 $L^{P_k}$ with weights tending to infinity when approaching points $a_k$, $b_k$.