Ten years ago I wrote about how cosine makes a decent approximation to the normal (Gaussian) probability density. It turns out you get a much better approximation if you raise cosine to a power.
If we normalize cosk(t) by dividing by its integral
we get an approximation to the density function for a normal distribution with mean 0 and variance 1/k.
Here are the plots of cosk(t), normalized to integrate to 1, for k = 1, 2, and 3.
And here’s a plot of cos3(t) compared to a normal with variance 1/3.
And finally here’s a plot of L² error, the square root of the integral of the square of the approximation error, as a function of k.
Update: You can do much better if you take a convex combination of cosine with 1 and allow non-integer powers. See this post.