Restore the signum of abs(sinc(x))

Question

Is it possible, by any means, to restore the signum of sinc(x) after being transformed to its absolute value, abs(sinc(x))? How it got to abs() is irrelevant, I only want to know if the reverse is possible.

Answer 1

Observe that $\operatorname{sinc }x=\frac{\sin x}{x}$ has its zeros at the nonzero integer multiples of $\pi$, and also observe that all of those zeros are simple. Thus knowing that $\operatorname{sinc x}$ is positive around $x=0$, we can determine the signum in the following way:

1. Given $x \in \mathbb{R}$ determine in which of the two families of intervals it lies: $\{[-\pi,\pi],[2\pi,3\pi],[-3 \pi,-2 \pi],\dots\}$ and $\{[\pi,2 \pi],[-2 \pi, -\pi],\dots \}$.

2. If it lies in the first family the signum is $+1$, otherwise it is $-1$.

If you feel like working a bit harder you can express the sign using the nearest integer function.

Answer 2

There is no way to restore the exact singum if you only know the absolute value, but in general the inverse of $y = |x|$ is $x = \pm y$