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For each and each positive integer , we give intrinsic characterizations of the restriction of the homogeneous Sobolev space to an arbitrary closed subset of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of spaces in the following sense: For every , it provides almost optimal -extensions of functions defined on . The operator norm of this extension operator is bounded by a constant depending only on . This enables us to prove several constructive -extension criteria expressed in terms of -th order divided differences of functions.
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Pavel Shvartsman, Extension criteria for homogeneous Sobolev spaces of functions of one variable. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 361–414DOI 10.4171/RMI/1210