If the unit square pulse $u_1(t)$ is defined as
$$u_0(t) = \begin{cases} 1 & \mathrm{if}\, 0 \le t \le 1 \\ 0 & \mathrm{otherwise} \end{cases}$$
and the function $u_n(t)$ is defined as
$$u_n(t) = (u_{n-1} \ast u_0)(t)$$
for integer $n > 0$, is there a closed-form solution for $u_n(t)$?
I know a couple of things:
- $u_1(t)$ is a triangular pulse with $u_1(0) = 0, u_1(1)=1, u_1(2) =0$
- $u_n(t)$ is a piecewise $n$th order polynomial with each nonzero piece occurring in the $n+1$ intervals $[0,1], [1,2], [2,3], \ldots [n,n+1]$
- as $n \to \infty$ the function becomes more and more of an approximation to a Gaussian function
- The Fourier transform of $u_n(t)$ has a closed-form expression as $U_n(\omega) = e^{j(n+1)\omega/2}\left(\frac{\sin(\omega/2)}{\omega/2}\right)^{n+1}$
but I'm not sure of much more than that. (I'm a bit rusty on my Fourier transforms so I may be off by a factor of 2 or $\pi$ or something.)