Recursive Convolutions of Unit Rectangle Function and Some Applications

Abstract

We study the sequence $f_n$ of functions that is defined by recursive convolutions as

$$\begin{aligned} f_1(x)= \Pi (x), \quad f_{n+1}(x)= (f_n *f_1)(x), \, n \in \mathbb {N}, \end{aligned}$$

where $\Pi$ is the unit rectangle function. We find out the general closed-form of the sequence $f_n$ and apply it for the evaluation of the improper integral

$$\begin{aligned} \frac{2}{\pi } \int _{0}^{\infty } \left( \frac{\sin (\xi )}{\xi }\right) ^n \cos (2 x \xi ) d\xi , \, x\in \mathbb {R},\, n \in \mathbb {N}, \, n\ge 2 . \end{aligned}$$

We also study some interesting features of the numerical coefficients that appear in the closed-form expression of $f_n$. In connection to the numerical coefficients that appear in closed-form expression of $f_n$, we introduce a map F defined on $\mathbb {N}\times \mathbb {N}_0$, by the rule, $F(n,s)=\sum _{i=0}^{n}\frac{i^s (-1)^{n-i}}{i!(n-i)!}$, and show that its range is in $\mathbb {N}_0$, where $\mathbb {N}_0:= \mathbb {N} \cup \{0\}$.