A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$ \int_{-\pi}^\pi |k_n(x)| \, \mathrm d x=O(1)$ where the implied constant is independent of $n$,
for each $\varepsilon>0$ we have $ \lim_{n\to \infty} \int_{\varepsilon<|x|\leq \pi} |k_n(x)| \, \mathrm d x=0$.
This is a standard definition in Fourier analysis, see, for example, the book of Stein--Shakarchi "Fourier analysis" page 48.
For example, if $$ k_n(x) = \sum_{k\in \mathbb Z \\ |k| \leq n} \exp(-\pi k^2/n^2) \mathrm e^{2\pi i k x}$$ then the integral in the third property has size roughly $\exp\left(-\pi \epsilon^2 n^2\right).$
QUESTION: is there a good choice for $k_n$ that makes the limit in $(3)$ be $$ o_{n\to +\infty }\left( \exp\left(-\pi \epsilon^2 n^2\right)\right)?$$
More importantly, is there any literature on this type of question?
