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Does someone have a "nice" example for a Sobolev function? I need a function in $W^{r, p}(0,1)$, besides the obvious ones like absolute value function or the squared absolute value function, preferable $r, p \in \{1, 2\}$, which I can use for a paper I write about the approximative properties of neural networks.

Edit: I have a theorem in my paper about the error approximation of Sobolev functions with neural networks. and I look at how the numerical results hold up compared to the theory. That's why I need a Sobolev function with discontinuity points, because a continious function would be to easy.

Lopsio
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    If $u\in W^{r,p}(0,1)$, and $F\in C^1(\mathbb R)$ then the composition $F\circ u$ is also in $W^{r,p}(0,1) $. So you can always compose your "obvious ones" with a more complicated $C^1$ function to create more examples. – Stratos supports the strike Oct 12 '23 at 13:09
  • Any $C^1$ function works ... $\sin$, $\cos$, $\exp$, polynomials ... please precise your question ... – LL 3.14 Oct 12 '23 at 16:06
  • $f = 0$ is an extremely interesting function in $W^{r,p}(0,1)$ :) – JayP Oct 13 '23 at 07:52

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