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Best approximation of L1 function by Lipschitz function

Let $(\rho_\epsilon)_{\epsilon>0}$ be a standard family of mollifiers, with $\rho_\epsilon$ supported in the ball $B_\epsilon(0)$. Since $\|\rho_\epsilon\|_{L^1}=1$ and $\|\rho_\epsilon\|_{L^\infty}=c …
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