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Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME

I apologize for repeating the same question from ME, but it seems more subtle than I expected. Let me fix the notations here first: \begin{equation} C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...