Balian-Low theorem

A theorem dealing with the representation of arbitrary functions in as a sum of time-frequency atoms, or Gabor functions (cf. also Gabor transform), of the form

where is a fixed window function and are fixed lattice parameters. The goal is to write an arbitrary function in a series of the form

(a1)

where the coefficients depend linearly on . One requires that the collection forms a frame for , that is, that there exist constants such that for any ,

(a2)

Inequality (a2) implies the existence of coefficients satisfying (a1) and the inequality . This inequality can be interpreted as expressing the continuous dependence of on the coefficients and the continuous dependence of these coefficients on . Whether or not an arbitrary collection of Gabor functions forms a frame for depends on the window function and on the lattice density . The lattice density is referred to as the critical density, for the following reason. If and forms a frame, then that frame is non-redundant, i.e., it is a Riesz basis. If and forms a frame, then that frame is redundant, i.e., the representation (a1) is not unique. If , then for any , the collection is incomplete. See [a9], [a10].

The time-frequency atom is said to be localized at time and frequency since the Fourier transform of is given by . A window function is said to have "good localization" in time and frequency if both and its Fourier transform decay rapidly at infinity. Good localization can be measured in various ways. One way is to require that . This is related to the classical uncertainty principle inequality, which asserts that any function satisfies .

The Balian–Low theorem asserts that if and if forms a frame for , then cannot have good localization. Specifically: If and if forms a frame for , then , i.e., 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, [a4], Thm. 3.2, in which a different criterion for "good localization" for the elements of a Gabor system is used, [a7], Thm. 4.4, in which more general time-frequency lattices for Gabor systems are considered, and [a3], which asserts a time-frequency restriction on bases of wavelets (cf. also Wavelet analysis).

The Balian–Low theorem was originally stated and proved by R. Balian [a1] and independently by F. Low [a8] under the stronger assumption that forms an orthonormal basis (cf. also Orthonormal system) for , and an extension of their argument to frames was given by I. Daubechies, R.R. Coifmann and S. Semmes [a5]. An elegant and entirely new proof of the theorem for orthonormal bases using the classical uncertainty principle inequality was given by G. Battle [a2], and an extension of this argument to frames was given by Daubechies and A.J.E.M. Janssen [a6]. Proofs of the Balian–Low theorem for frames use the differentiability properties of the Zak transform in an essential way.

References

[a1]	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
[a2]	G. Battle, "Heisenberg proof of the Balian–Low theorem" Lett. Math. Phys. , 15 (1988) pp. 175–177
[a3]	G. Battle, "Phase space localization theorem for ondelettes" J. Math. Phys. , 30 (1989) pp. 2195–2196
[a4]	J. Benedetto, C. Heil, D. Walnut, "Differentiation and the Balian–Low Theorem" J. Fourier Anal. Appl. , 1 (1995) pp. 355–402
[a5]	I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis" IEEE Trans. Inform. Th. , 39 (1990) pp. 961–1005
[a6]	I. Daubechies, A.J.E.M. Janssen, "Two theorems on lattice expansions" IEEE Trans. Inform. Th. , 39 (1993) pp. 3–6
[a7]	H. Feichtinger, K. Gröchenig, "Gabor frames and time—frequency distributions" J. Funct. Anal. , 146 (1997) pp. 464–495
[a8]	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
[a9]	M. Rieffel, "Von Neumann algebras associated with pairs of lattices in Lie groups" Math. Ann. , 257 (1981) pp. 403–418
[a10]	J. Ramanathan, T. Steger, "Incompleteness of Sparse Coherent States" Appl. Comput. Harm. Anal. , 2 (1995) pp. 148–153