View source for Balian-Low theorem

A theorem dealing with the representation of arbitrary functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200301.png" /> as a sum of time-frequency atoms, or Gabor functions (cf. also [[Gabor transform|Gabor transform]]), of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200302.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200303.png" /> is a fixed window function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200304.png" /> are fixed lattice parameters. The goal is to write an arbitrary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200305.png" /> in a series of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200307.png" /> depend linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200308.png" />. One requires that the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b1200309.png" /> forms a frame for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003010.png" />, that is, that there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003011.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003012.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Inequality (a2) implies the existence of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003014.png" /> satisfying (a1) and the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003015.png" />. This inequality can be interpreted as expressing the continuous dependence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003016.png" /> on the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003017.png" /> and the continuous dependence of these coefficients on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003018.png" />. Whether or not an arbitrary collection of Gabor functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003019.png" /> forms a frame for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003020.png" /> depends on the window function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003021.png" /> and on the lattice density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003022.png" />. The lattice density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003023.png" /> is referred to as the critical density, for the following reason. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003025.png" /> forms a frame, then that frame is non-redundant, i.e., it is a [[Riesz basis|Riesz basis]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003027.png" /> forms a frame, then that frame is redundant, i.e., the representation (a1) is not unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003028.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003029.png" />, the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003030.png" /> is incomplete. See [[#References|[a9]]], [[#References|[a10]]].

The time-frequency atom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003031.png" /> is said to be localized at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003032.png" /> and frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003033.png" /> since the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003034.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003035.png" />. A window function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003036.png" /> is said to have  "good localization"  in time and frequency if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003037.png" /> and its Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003038.png" /> decay rapidly at infinity. Good localization can be measured in various ways. One way is to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003039.png" />. This is related to the classical uncertainty principle inequality, which asserts that any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003040.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003041.png" />.

The Balian–Low theorem asserts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003042.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003043.png" /> forms a frame for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003045.png" /> cannot have good localization. Specifically: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003046.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003047.png" /> forms a frame for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003049.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003050.png" /> maximizes the uncertainty principle inequality.

More generally, the term  "Balian–Low theorem"  or  "Balian–Low-type theorem"  can refer to any theorem which asserts time and frequency localization restrictions on the elements of a Riesz basis. Such theorems include, for example, [[#References|[a4]]], Thm. 3.2, in which a different criterion for  "good localization"  for the elements of a Gabor system is used, [[#References|[a7]]], Thm. 4.4, in which more general time-frequency lattices for Gabor systems are considered, and [[#References|[a3]]], which asserts a time-frequency restriction on bases of wavelets (cf. also [[Wavelet analysis|Wavelet analysis]]).

The Balian–Low theorem was originally stated and proved by R. Balian [[#References|[a1]]] and independently by F. Low [[#References|[a8]]] under the stronger assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003051.png" /> forms an orthonormal basis (cf. also [[Orthonormal system|Orthonormal system]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120030/b12003052.png" />, and an extension of their argument to frames was given by I. Daubechies, R.R. Coifmann and S. Semmes [[#References|[a5]]]. An elegant and entirely new proof of the theorem for orthonormal bases using the classical uncertainty principle inequality was given by G. Battle [[#References|[a2]]], and an extension of this argument to frames was given by Daubechies and A.J.E.M. Janssen [[#References|[a6]]]. Proofs of the Balian–Low theorem for frames use the differentiability properties of the Zak transform in an essential way.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Balian,   "Un principe d'incertitude fort en théorie du signal ou en mécanique quantique"  ''C.R. Acad. Sci. Paris'' , '''292'''  (1981)  pp. 1357–1362</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Battle,   "Heisenberg proof of the Balian–Low theorem"  ''Lett. Math. Phys.'' , '''15'''  (1988)  pp. 175–177</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Battle,   "Phase space localization theorem for ondelettes"  ''J. Math. Phys.'' , '''30'''  (1989)  pp. 2195–2196</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Benedetto,   C. Heil,   D. Walnut,   "Differentiation and the Balian–Low Theorem"  ''J. Fourier Anal. Appl.'' , '''1'''  (1995)  pp. 355–402</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Daubechies,   "The wavelet transform, time-frequency localization and signal analysis"  ''IEEE Trans. Inform. Th.'' , '''39'''  (1990)  pp. 961–1005</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Daubechies,   A.J.E.M. Janssen,   "Two theorems on lattice expansions"  ''IEEE Trans. Inform. Th.'' , '''39'''  (1993)  pp. 3–6</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Feichtinger,   K. Gröchenig,   "Gabor frames and time—frequency distributions"  ''J. Funct. Anal.'' , '''146'''  (1997)  pp. 464–495</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  F. Low,   "Complete sets of wave packets"  C. DeTar (ed.)  et al. (ed.) , ''A Passion for Physics: Essays in Honor of Geoffrey Chew'' , World Sci.  (1985)  pp. 17–22</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Rieffel,   "Von Neumann algebras associated with pairs of lattices in Lie groups"  ''Math. Ann.'' , '''257'''  (1981)  pp. 403–418</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Ramanathan,   T. Steger,   "Incompleteness of Sparse Coherent States"  ''Appl. Comput. Harm. Anal.'' , '''2'''  (1995)  pp. 148–153</TD></TR></table>