I'm studying Fourier analysis and have a question about approximate identities.


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（not necessarily $f$） everywhere ? 


I have asked this question on math.stackexchange but I didn't get an answer.



Approximate identities and pointwise convergence
No. Given any $k\_{\epsilon}$, we can easily build an $f$ so that $I(\epsilon):=k\_{\epsilon}\*f(0)$ oscillates roughly between $-1$ and $1$ as $\epsilon\to 0$. My $f$ will only take the two values $\pm 1$.


Just take $f=1$ everywhere except on a tiny interval centered at $0$, and take this interval so small that $I(1)\simeq 1$. Next, take $\epsilon\_2\ll 1$ so small that $k\_{\epsilon\_2}$ is essentially supported by that tiny interval that is left, and let $f=-1$ there, except on an even smaller interval centered at $0$. Again, this will be chosen so small that $I(\epsilon\_2)\simeq -1$ etc.


