Editing Uncertainty principle (section)

=== Hardy's uncertainty principle ===
The mathematician [[G. H. Hardy]] formulated the following uncertainty principle:<ref>{{Citation|first=G.H.|last=Hardy|author-link=G. H. Hardy|title=A theorem concerning Fourier transforms|journal=Journal of the London Mathematical Society|volume=8|year=1933|issue=3|pages=227–231|doi=10.1112/jlms/s1-8.3.227}}</ref> it is not possible for {{mvar|f}} and {{math| ƒ̂}} to both be "very rapidly decreasing". Specifically, if {{mvar|f}} in <math>L^2(\mathbb{R})</math> is such that
<math display="block">|f(x)|\leq C(1+|x|)^Ne^{-a\pi x^2}</math>
and
<math display="block">|\hat{f}(\xi)|\leq C(1+|\xi|)^Ne^{-b\pi \xi^2}</math> (<math>C>0,N</math> an integer),
then, if {{math|1=''ab'' > 1, ''f'' = 0}}, while if {{math|1=''ab'' = 1}}, then there is a polynomial {{mvar|P}} of degree {{math|≤ ''N''}} such that
<math display="block">f(x)=P(x)e^{-a\pi x^2}. </math>

This was later improved as follows: if <math>f \in L^2(\mathbb{R}^d)</math> is such that
<math display="block">\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}|f(x)||\hat{f}(\xi)|\frac{e^{\pi|\langle x,\xi\rangle|}}{(1+|x|+|\xi|)^N} \, dx \, d\xi < +\infty ~,</math>
then
<math display="block">f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
where {{mvar|P}} is a polynomial of degree {{math|(''N'' − ''d'')/2}} and {{mvar|A}} is a real {{math|''d'' × ''d''}} positive definite matrix.

This result was stated in Beurling's complete works without proof and proved in Hörmander<ref>{{Citation | first=L. | last=Hörmander | author-link=Lars Hörmander|title=A uniqueness theorem of Beurling for Fourier transform pairs|journal= Ark. Mat. | volume=29|issue=1–2|year=1991|pages=231–240|bibcode=1991ArM....29..237H|doi=10.1007/BF02384339|s2cid=121375111 | doi-access=free}}</ref> (the case <math>d=1,N=0</math>) and Bonami, Demange, and Jaming<ref>{{Citation | first1=A. | last1=Bonami | author1-link= Aline Bonami |first2=B.|last2=Demange|first3=Ph.|last3=Jaming|title=Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms |journal= Rev. Mat. Iberoamericana | volume=19 | year=2003 | pages=23–55 | bibcode=2001math......2111B|arxiv=math/0102111| doi=10.4171/RMI/337|s2cid=1211391}}</ref> for the general case. Note that Hörmander–Beurling's version implies the case {{math|''ab'' > 1}} in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in ref.<ref>{{Citation|first=Haakan|last=Hedenmalm|title=Heisenberg's uncertainty principle in the sense of Beurling|journal=[[Journal d'Analyse Mathématique]] | volume=118 | issue=2 | year=2012 | pages=691–702 | doi=10.1007/s11854-012-0048-9 | doi-access=free | arxiv=1203.5222 | bibcode=2012arXiv1203.5222H | s2cid=54533890}}</ref>

A full description of the case {{math|''ab'' < 1}} as well as the following extension to Schwartz class distributions appears in ref.<ref>{{Citation|first=Bruno|last=Demange|title=Uncertainty Principles Associated to Non-degenerate Quadratic Forms|year=2009|publisher= Société Mathématique de France|isbn=978-2-85629-297-6}}</ref>

{{math theorem| If a tempered distribution <math>f\in\mathcal{S}'(\R^d)</math> is such that
<math display="block">e^{\pi|x|^2}f\in\mathcal{S} '(\R^d)</math>
and
<math display="block">e^{\pi|\xi|^2}\hat f\in\mathcal{S}'(\R^d) ~,</math>
then
<math display="block">f(x)=P(x)e^{-\pi\langle Ax,x\rangle} ~,</math>
for some convenient polynomial {{mvar|P}} and real positive definite matrix {{mvar|A}} of type {{math|''d'' × ''d''}}.}}