Are good kernels trigonometric polynomials?

Question

We know :
Definition: A kernel $K_n$ is 'good' if they are integrable and satisfy the following conditions:

1. $\int_{-\pi}^{\pi}K_n(x)dx=1$
2. $\int_{-\pi}^{\pi}|K_n(x)|dx\le A$ for some $A>0$
3. For every $\eta>0$, $$\int_{|x|\ge\eta}|K_n(x)|dx\to0\text{ }\text{ }\text{ as } n\to0$$

My question is Are all good kernals trigonometric polynomials, and if so:
Is the convolution of a good kernal with a $2\pi$ periodic function a trigonometric polynomial?