  We propose new repair schemes for Reed-Solomon codes that use subspace
polynomials and hence generalize previous works in the literature that employ
trace polynomials. The Reed-Solomon codes are over $\mathbb{F}_{q^\ell}$ and
have redundancy $r = n-k \geq q^m$, $1\leq m\leq \ell$, where $n$ and $k$ are
the code length and dimension, respectively. In particular, for one erasure, we
show that our schemes can achieve optimal repair bandwidths whenever $n=q^\ell$
and $r = q^m,$ for all $1 \leq m \leq \ell$. For two erasures, our schemes use
the same bandwidth per erasure as the single erasure schemes, for $\ell/m$ is a
power of $q$, and for $\ell=q^a$, $m=q^b-1>1$ ($a \geq b \geq 1$), and for
$m\geq \ell/2$ when $\ell$ is even and $q$ is a power of two.
