In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula is accurate?
Also, what is the formula for the maximum value of multiple self-convolution of rectangular function?
I'll just address your first question.
Let $$ h(t) = \begin{matrix} 1 &\; \text{if} \; t \in [0,1] \\ 0 &\; \text{else}\end{matrix} $$ and let $$ g_n(t) = (h*h*\cdots*h)(t) $$ be the n-fold convolution of $h$. Observe that $$ \widehat{g_n}(w) = \left(\frac{1}{iw}\right)^n\left(1-e^{-iw}\right)^n =: \hat{b}(w)\hat{c}(w) $$ where $\hat{f}(w) = \int_{\mathbb{R}} f(t) e^{-iwt} dt$. So then by the convolution theorem, we have $$ g_n(t) = (b*c)(t) $$
To find $b(t)$ and $c(t)$, you will have to find these distributional inverse transforms, and use the binomial theorem for $\hat{c}(w)$. I'll leave this to you. You should find that:
$$ g_n(t) = \frac{1}{2(n-1)!}\sum_{j=0}^n \begin{pmatrix} n \\ j \end{pmatrix} (-1)^j (t-j)^{n-1} \text{sign}(t-j) $$
You of course would have to show this is equivalent to your linked question.
