A Simple Observation on the Uncertainty Principle for the Fractional Fourier Transform

The aim of this letter is to show that uncertainty principles for the pair $$u,{\mathcal F}[u]$$ ( $${\mathcal F}$$ the Fourier transform) can be transferred without much effort to the pair $${\mathcal F}_\alpha [u],{\mathcal F}_\beta [u]$$ ( $${\mathcal F}_\alpha $$ the Fractional Fourier transform) provided $$\beta -\alpha \notin \pi {\mathbb {Z}}$$ . This letter is essentially of a tutorial nature and aims at avoiding that people waste their efforts on adaptations of the proofs from classical Fourier analysis to the fractional setting.