Editing Uncertainty principle (section)

==Harmonic analysis==
{{Main article|Fourier transform#Uncertainty principle}}
In the context of [[harmonic analysis]] the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds,
<math display="block">\left(\int_{-\infty}^\infty x^2 |f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty \xi^2 |\hat{f}(\xi)|^2\,d\xi\right)\ge \frac{\|f\|_2^4}{16\pi^2}.</math>

Further mathematical uncertainty inequalities, including the above [[entropic uncertainty]], hold between a function {{mvar|f}} and its Fourier transform {{math| ƒ̂}}:<ref>{{Citation|first1=V.|last1=Havin|first2= B.|last2=Jöricke|title=The Uncertainty Principle in Harmonic Analysis|publisher=Springer-Verlag|year=1994}}</ref><ref>{{Citation | last1 = Folland | first1 = Gerald | last2 = Sitaram |first2 = Alladi | title = The Uncertainty Principle: A Mathematical Survey | journal = Journal of Fourier Analysis and Applications | date = May 1997 | volume = 3 | issue = 3 | pages = 207–238 | doi = 10.1007/BF02649110 | bibcode = 1997JFAA....3..207F | mr=1448337 | s2cid = 121355943 }}</ref><ref>{{springer|title=Uncertainty principle, mathematical|id=U/u130020|first=A|last=Sitaram|year=2001}}</ref>
<math display="block">H_x+H_\xi \ge \log(e/2)</math>

===Signal processing {{anchor|Gabor limit}}===
In the context of [[time–frequency analysis]] uncertainty principles are referred to as the '''Gabor limit''', after [[Dennis Gabor]], or sometimes the ''Heisenberg–Gabor limit''. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both [[time limited]] and [[band limited]] (a function and its Fourier transform cannot both have bounded domain)—see [[Bandlimiting#Bandlimited versus timelimited|bandlimited versus timelimited]]. More accurately, the ''time-bandwidth'' or ''duration-bandwidth'' product satisfies
<math display="block">\sigma_{t} \sigma_{f} \ge \frac{1}{4\pi} \approx 0.08 \text{ cycles},</math>
where <math>\sigma_{t}</math> and <math>\sigma_{f}</math> are the standard deviations of the time and frequency energy concentrations respectively.<ref>{{cite book | last=Mallat | first=S. G. | title=A wavelet tour of signal processing: the sparse way | publisher=Elsevier/Academic Press | publication-place=Amsterdam ; Boston | date=2009 | isbn=978-0-12-374370-1|doi=10.1016/B978-0-12-374370-1.X0001-8|page=44}}</ref> The minimum is attained for a [[Gaussian function|Gaussian]]-shaped pulse ([[Gabor wavelet]]) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitude <math>\sigma_t\sigma_f=1/2\pi</math>; squaring reduces each <math>\sigma</math> by a factor <math>\sqrt 2</math>.] Another common measure is the product of the time and frequency [[full width at half maximum]] (of the power/energy), which for the Gaussian equals <math>2 \ln 2 / \pi \approx 0.44</math> (see [[bandwidth-limited pulse]]).

Stated differently, one cannot simultaneously sharply localize a signal {{mvar|f}} in both the [[time domain]] and [[frequency domain]].

When applied to [[Filter (signal processing)|filters]], the result implies that one cannot simultaneously achieve a high temporal resolution and high frequency resolution at the same time; a concrete example are the [[Short-time Fourier transform#Resolution issues|resolution issues of the short-time Fourier transform]]—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off.

Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice, the Gabor limit limits the ''simultaneous'' time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.

As a result, in order to analyze signals where the [[Transient (acoustics)|transients]] are important, the [[Wavelet Transform|wavelet transform]] is often used instead of the Fourier.

===Discrete Fourier transform===
Let <math>\left \{ \mathbf{ x_n } \right \} := x_0, x_1, \ldots, x_{N-1}</math> be a sequence of ''N'' complex numbers and <math>\left \{ \mathbf{X_k} \right \} := X_0, X_1, \ldots, X_{N-1},</math> be its [[Discrete Fourier transform#Uncertainty principles | discrete Fourier transform]].

Denote by <math>\|x\|_0</math> the number of non-zero elements in the time sequence <math>x_0,x_1,\ldots,x_{N-1}</math> and by <math>\|X\|_0</math> the number of non-zero elements in the frequency sequence <math>X_0,X_1,\ldots,X_{N-1}</math>. Then,
<math display="block">\|x\|_0 \cdot \|X\|_0 \ge N.</math>

This inequality is [[inequality (mathematics)#Sharp inequalities|sharp]], with equality achieved when ''x'' or ''X'' is a Dirac mass, or more generally when ''x'' is a nonzero multiple of a Dirac comb supported on a subgroup of the integers modulo ''N'' (in which case ''X'' is also a Dirac comb supported on a complementary subgroup, and vice versa).

More generally, if ''T'' and ''W'' are subsets of the integers modulo ''N'', let <math>L_T,R_W : \ell^2(\mathbb Z/N\mathbb Z)\to\ell^2(\mathbb Z/N\mathbb Z)</math> denote the time-limiting operator and [[bandlimiting|band-limiting operator]]s, respectively. Then
<math display="block">\|L_TR_W\|^2 \le \frac{|T||W|}{|G|} </math>
where the norm is the [[operator norm]] of operators on the Hilbert space <math>\ell^2(\mathbb Z/N\mathbb Z)</math> of functions on the integers modulo ''N''. This inequality has implications for [[signal reconstruction]].<ref name="Donoho">{{cite journal |last1=Donoho |first1=D.L. |last2=Stark |first2=P.B |year=1989 |title=Uncertainty principles and signal recovery |journal=SIAM Journal on Applied Mathematics |volume=49 |issue=3 |pages=906–931 |doi=10.1137/0149053}}</ref>

When ''N'' is a [[prime number]], a stronger inequality holds:
<math display="block">\|x\|_0 + \|X\|_0 \ge N + 1.</math>
Discovered by [[Terence Tao]], this inequality is also sharp.<ref>{{citation|
journal=Mathematical Research Letters
|volume=12
|year=2005
|issue=1
|title=An uncertainty principle for cyclic groups of prime order
|pages=121–127
|author=[[Terence Tao]]
|doi=10.4310/MRL.2005.v12.n1.a11
|arxiv=math/0308286
|s2cid=8548232
}}</ref>

=== 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.

=== 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''}}.}}