  For each $p>1$ and each positive integer $m$ we give intrinsic
characterizations of the restriction of the homogeneous Sobolev space
$L^m_p(R)$ to an arbitrary closed subset $E$ of the real line.
  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 $L^m_p$-extension criteria expressed in terms of $m^{th}$ order
divided differences of functions.
