A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that $$ \int_0^1 k_n(t) \mathrm d t =1,$$ $$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant independent of $n$ and, lastly, $k_n$ must satisfy that for every $\delta \in (0,1/2]$, $$ \lim_{n\to+\infty}\int_{\delta<|t|<1/2} |k_n(t)| \mathrm d t =0.$$
My question is whether we can choose a summability kernel such that the limit in the third property converges faster than the Fejér kernel. Specifically, the Fejér kernel is defined as $$F_n(t) = \frac{1}{n}\sum_{|k|\leq n-1} \left(1-\frac{|k|}{n}\right)\mathrm e^{2\pi i k t}$$ and one can show that it is given in closed form as $$F_n(t)=\frac{1}{n}\frac{1-\cos(n t) }{1-\cos t}.$$ This is $\ll \frac{1}{n|1-\cos t|}$, which is $\ll 1/(n t^2)$ from the Taylor expansion of $\cos$ around $t=0$. Hence $$\int_{\delta<|t|<1/2} |k_n(t)| \mathrm d t \asymp \frac{1}{n \delta}.$$ This goes to zero as long as $$\delta>\frac{1}{n^{1.0001}},$$ say.
QUESTION: is there any other kernel such that the limit goes to zero even when $\delta $ is not so large compared to $n$? For example, when $$\delta>\frac{1}{n^c}$$ for some constant $0<c<1$ ?
