For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset of the real line.
Sobolev $W^1_p$-spaces on closed subsets of $R^n$
Sobolev functions on closed subsets of the real line: long version
Extension criterions for homogeneous Sobolev space of functions of one variable
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