I'm studying Fourier analysis and learned the concept of approximate identity.
Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $\epsilon\to 0$.
Question: Can we construct a $k_{\epsilon}$ such that for every $f\in L^{1}$, $k_{\epsilon}*f$ converges everywhere ?