The Dirichlet kernels are defined by

D_k(x)= \frac{\sin((k+1/2)x)}{2\pi \sin(x/2)}

Here is an animation of the first 15 Dirichlet kernels.

The Fejer kernels are defined by

F_k(x)=\frac{\sin(kx/2)^2}{2\pi k \sin(x/2)^2}

Here's an animation of the first 15 Fejer kernels.

The Dirichlet and Fejer kernels are periodic functions. If we look at plots of them on larger windows, then the graphs repeat. These are plots of the 15th Dirichlet and Fejer kernels.

         

Another sequence of approximate delta functions appear when we study the Fourier transform.

K_r(x)= \frac{\sin(r x)}{\pi x}

The functions K_r(x) come up somewhat often, but as far as I know don't have a special name. The variable r is not necessarily integer valued. Notice that in contrast to the Dirichlet and Fejer kernels, these functions are not periodic. Here is an animation with r going from 1 to 20, advancing in increments of 0.3 between frames.