Uncertainty principles in Fourier analysis

Uncertainty principles in Fourier analysis are formalizations of the following principle: A function $f$ and its Fourier transform $\hat{f}$ cannot both be sharply ''localized''. This of course allows many interpretations, and by interpreting localization in various ways, we get theorems (usually inequalities) about the simultaneous behavior of $f$ and $\hat{f}$. We will formalize and prove the following uncertainty principles.

• $|f|^2$ and $|\hat{f}|^2$ cannot both have small variance (Heisenberg uncertainty principle)
•  Two self-adjoint operators cannot both have small norms compared to the norm of their commutator (Heisenberg uncertainty principle for operators)
•  $f$ and $\hat{f}$ cannot both decay faster than gaussian functions (Hardy's uncertainty principle)
• Not both $f$ and $\hat{f}$ can be nonzero only in a set of finite measure (Benedick's inequality)
• The entropy of $f$ and $\hat{f}$ cannot both be positive (Hirschman's inequality)

The first hint that such principles might be true is that the Fourier transform ''spreads out'' a function the more it is concentrated: $\mathcal{F}(f(tx))=\frac{1}{t}\hat{f}(\frac{\xi}{t})$. We now proceed to formalize and prove the principles.

Fourier transform and its mapping properties

We present some classical results about the Fourier transform in $\mathbb{R}^n$ in this post, and some of them will be applied in later posts to partial differential equations, additive number theory, and other topics. In particular, we consider the validity of the Fourier inversion theorem in various forms, the $L^2$ theory of the Fourier transform and the mapping properties of the Fourier transform in many spaces.