Editing Uncertainty principle (section)

=== Benedicks's theorem ===
Amrein–Berthier<ref>
{{citation
 | last1 = Amrein | first1 = W.O.
 | last2 = Berthier | first2 = A.M.
 | year = 1977
 | title = On support properties of ''L''<sup>''p''</sup>-functions and their Fourier transforms
 | journal = Journal of Functional Analysis
 | volume = 24 | issue = 3 | pages = 258–267
 | doi = 10.1016/0022-1236(77)90056-8
 | postscript = .
 | doi-access = free
}}</ref> and Benedicks's theorem<ref>{{citation |first=M. |last=Benedicks |author-link=Michael Benedicks |title=On Fourier transforms of functions supported on sets of finite Lebesgue measure |journal=J. Math. Anal. Appl. |volume=106 |year=1985 |issue=1 |pages=180–183 |doi=10.1016/0022-247X(85)90140-4 |doi-access=free }}</ref> intuitively says that the set of points where {{mvar|f}} is non-zero and the set of points where {{math|ƒ̂}} is non-zero cannot both be small.

Specifically, it is impossible for a function {{mvar|f}} in {{math|''L''<sup>2</sup>('''R''')}} and its Fourier transform {{math|ƒ̂}} to both be [[support of a function|supported]] on sets of finite [[Lebesgue measure]]. A more quantitative version is<ref>{{Citation|first=F.|last=Nazarov|author-link=Fedor Nazarov|title=Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type|journal=St. Petersburg Math. J.|volume=5|year=1994|pages=663–717}}</ref><ref>{{Citation|first=Ph.|last=Jaming|title=Nazarov's uncertainty principles in higher dimension|journal= J. Approx. Theory|volume=149|year=2007|issue=1|pages=30–41|doi=10.1016/j.jat.2007.04.005|arxiv=math/0612367|s2cid=9794547}}</ref>
<math display="block">\|f\|_{L^2(\mathbf{R}^d)}\leq Ce^{C|S||\Sigma|} \bigl(\|f\|_{L^2(S^c)} + \| \hat{f} \|_{L^2(\Sigma^c)} \bigr) ~.</math>

One expects that the factor {{math|''Ce''<sup>''C''{{abs|''S''}}{{abs|''Σ''}}</sup>}} may be replaced by {{math|''Ce''<sup>''C''({{abs|''S''}}{{abs|''Σ''}})<sup>1/''d''</sup></sup>}}, which is only known if either {{mvar|S}} or {{mvar|Σ}} is convex.