  For each $p>1$ and each positive integer $m$ we give intrinsic
characterizations of the restriction of the Sobolev space $W^m_p(R)$ and
homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the
real line. In particular, we show that the classical one dimensional Whitney
extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the
following sense: for every $p\in(1,\infty]$ it provides almost optimal
$L^m_p$-extensions of functions defined on $E$. The operator norm of this
extension operator is bounded by a constant depending only on $m$. This enables
us to prove several constructive $W^m_p$- and $L^m_p$-extension criteria
expressed in terms of $m^{th}$ order divided differences of functions.
