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.

# Fourier Transform

# Fourier Transform over Finite Abelian Groups

Here is an overview of Fourier analysis applied to finite abelian groups. It shows how to generalize a priori different computations such as the discrete Fourier transform or the Hadamard-Walsh transform.

**CHARACTER**

Let be a finite abelian group with additive notation. A character of is a group homomorphism that is to say for all