Heisenberg inequality on the real line

A quantum mechanical principle discovered by Werner Heisenberg states that it is not possible to simultaneously determine the position and momentum of a particle. In fact, this principle is deeply mathematical and independent from the experimental precision of the instruments. We illustrate how it can be expressed for functions on the real line.
Let $G$ be a finite abelian group with additive notation. A character of $G$ is a group homomorphism $\chi : G \rightarrow \mathbb{C} - \{ 0 \}$ that is to say for all $a , b \in G$
$\displaystyle \chi ( a + b ) = \chi ( a ) \chi ( b )$
$\displaystyle \chi ( 0 ) = 1$