Convergence of Poisson integral at boundary of half plane.

Question

My question is given $f\in L^1(\mathbb{R})$ and defining $$ f_{\epsilon}(u)=\frac{1}{\pi}\int_{\mathbb{R}}f(t)\frac{\epsilon}{(t-u)^2+\epsilon^2}dt, $$ whether $f_\epsilon$ converges to $f$ in $L^1$? If not, then is it true if $f$ has compact support or for $L^p$ spaces with $p>1$?