For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R}$ to an arbitrary closed subset of the real line.
Sobolev functions on closed subsets of the real line
Sobolev $W^1_p$-spaces on closed subsets of $R^n$
Densely Defined Multiplication on the Sobolev Space
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