I'm self-studying math, and came across a problem:
$$ \prod_{n=0}^{N-1} \left(u[x_n + \theta] - u[x_n-\theta]\right) = u[\theta - \max(|x_n| )] $$
where $u$ is a unit step function, $x_n$ are sample points, $\theta$ is a constant. So basically, the two unit steps functions form a rectangular function, and the centers $x_n$ of the rectangular functions are different and if they don't overlap then the whole product is zero. And we are to find how to have this product not equal to zero. So by the explanation from the book, if $-\theta < x[n] < \theta$, this product equals to $u(\theta - \max(|x[n]| )$. Why is that?
The step function $u[t]$ equals $1$ if its argument $t$ is greater than zero, and equals $0$ otherwise. Another way to write this is $$u[t] = I(t>0),\tag1$$ where $I(\cdot)$ denotes the indicator function, which is one when its argument is true and zero otherwise. Your book argues that the expression $$\prod_{n=0}^{N-1} \left(u[x_n + \theta] - u[x_n-\theta]\right)\tag2$$ equals one if $\max|x_n|<\theta$, and equals zero otherwise. This means expression (2) is equal to $$I(\max|x_n|<\theta),$$ which can be rewritten $I(\theta-\max|x_n|>0)$, which by (1) can be rewritten $u[\theta-\max|x_n|]$.