A class of functions on a disjoint collection of segments

Let $E=\bigcup\limits^m_{k=1}S_k$, where $S_k$ are disjoint segments, and let $\{\alpha_k\}$ be a collection of positive numbers, $0<\alpha_k<1$. We describe a class of functions $f$ on $E$ that admit approximation by polynomials of degree $\le n$ with the rate $\frac1{n^{\alpha_k}}$ on $S_k$.