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

Sobolev functions on closed subsets of the real line: long version