Can cosine kernel be understood as a case of Beta distribution? As noted by Wand and Jones (1995), most standard kernels may be seen as a case of
$$ K(x;p) = \{ 2^{2p+1} \; \mathrm{B}(p+1,p+1) \}^{-1} \; (1-x^2)^p \;\boldsymbol{1}_{\{|x|<1\}} $$
family, where $\mathrm{B}(\cdot,\cdot)$ is a Beta function. Different values of $p$ lead to rectangular ($p=0$), Epanechnikov ($p=1$), biweight ($p=2$) and triweight ($p=3$) kernels.
Can cosine kernel (as understood in R's density function),
$$ \frac{1}{2} (1 + \cos(\pi x)) \;\boldsymbol{1}_{\{|x|<1\}} $$
also be thought as a member of this family? If so, what is an appropriate value of $p$ for it? After doing some simulations I guess that $\approx 2.35$ is pretty close, but (how) can I find the proper without simulation? If not, can it be approximated using beta distribution?

Wand, M. P. and Jones, M. C. (1995).
Kernel Smoothing.
Chapman and Hall, London.
 A: The cosine kernel is not a beta distribution.
Note that the following things are all true of the standard cosine density:


*

*$f(0)=1$

*$f(0.5)=0.5$

*The right half of this density is rotationally symmetric about $x=\frac12$: (i.e. considering the other two properties it implies $1-f(x)=f(1-x)$ )
But no beta density on (-1,1) will have all
 these properties together.
The symmetric beta kernel density can be written as:
$g(x;a)= \frac{(1-x^2)^{a-1}}{\text{B}(a,a)2^{2a-1}}\,,\:-1<x<1\,,\:a>0$
For example, the first condition implies a $a$ of about $3.38175$ ($p=2.38175$). The second implies an $a$ of 1 ($p=0$).
However, values of $a$ near that choice of $a$ (3.38175) gives densities really quite close to the cosine.
[This is quite close to your $p=2.35$ (since $p=a-1$); a range of values in this region give densities similar to the cosine.] 
The smallest absolute deviation in density happens for $p\approx 2.3575$ -- not that minimizing the absolute deviations will make the properties most alike.
Here's the cosine and beta (with $p=2.3575$):

Even though they're not the same, they're quite alike in shape.
