Background Given a sequence ${K_n}$ of functions on region $T$. Then ${K_n}$ is a good kernel if:
(1) $\frac{1}{2\pi}\int_{-\pi}^{\pi} K_{n}(x)dx = 1$ $\forall n \in N$
(2) $\frac{1}{2\pi}\int_{-\pi}^{\pi} |K_{n}(x)|dx = O(1)$
(3) If $\delta \in (0,\pi)$, $\int_{\delta \leq |x|\leq \pi} |K_{n}(x)|\ dx \rightarrow 0$ as $\ n\rightarrow \infty$
Problem I am trying to come up with an example of a NOT good kernel which will satisfy $(1)$ and $(2)$ but fails to satisfy $(3)$. But I only tried to mimic the construction of Dirichlet Kernel to see if I can find such not good kernel, but unfortunately nothing is successful. Can anyone please help on this problem?