Balian-Low theorem

A theorem dealing with the representation of arbitrary functions in $L ^ { 2 } ( \mathbf{R} )$ as a sum of time-frequency atoms, or Gabor functions (cf. also Gabor transform), of the form

\begin{equation*} \{ e ^ { 2 \pi i m b x } g ( x - n a ) : n , m \in \mathbf{Z} \} = \{ g _ { n , m} : n , m \in \mathbf{Z} \}, \end{equation*}

where $g \in L ^ { 2 } ( \mathbf{R} )$ is a fixed window function and $a , b > 0$ are fixed lattice parameters. The goal is to write an arbitrary function $f \in L ^ { 2 } ( \mathbf{R} )$ in a series of the form

\begin{equation} \tag{a1} f ( x ) = \sum _ { n \in \mathbf{Z} } \sum _ { m \in \mathbf{Z} } c _ { n , m } (\, f ) g _ { n , m } ( x ), \end{equation}

where the coefficients $\{ c _ { n ,m} ( f ) : n , m \in \mathbf{Z} \}$ depend linearly on $f$. One requires that the collection $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame for $L ^ { 2 } ( \mathbf{R} )$, that is, that there exist constants $A , B > 0$ such that for any $f \in L ^ { 2 } ( \mathbf{R} )$,

\begin{equation} \tag{a2} A \| f \| _ { 2 } ^ { 2 } \leq \sum _ { n \in \mathbf Z } \sum _ { m \in \mathbf Z } |\langle f , g _ { n , m} \rangle | ^ { 2 } \leq B \| f \| _ { 2 } ^ { 2 }. \end{equation}

Inequality (a2) implies the existence of coefficients $\{ c _ { n ,m} ( f ) : n , m \in \mathbf{Z} \}$ satisfying (a1) and the inequality $B ^ { - 1 } \| f \| _ { 2 } ^ { 2 } \leq \sum _ { n , m \in \mathbf{Z} } | c _ { n , m } ( f ) | ^ { 2 } \leq A ^ { - 1 } \| f \| _ { 2 } ^ { 2 }$. This inequality can be interpreted as expressing the continuous dependence of $f$ on the coefficients $\{ c _ { n ,m} ( f ) : n , m \in \mathbf{Z} \}$ and the continuous dependence of these coefficients on $f$. Whether or not an arbitrary collection of Gabor functions $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame for $L ^ { 2 } ( \mathbf{R} )$ depends on the window function $g$ and on the lattice density $( a b ) ^ { - 1 }$. The lattice density $( a b ) ^ { - 1 } = 1$ is referred to as the critical density, for the following reason. If $( a b ) ^ { - 1 } = 1$ and $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame, then that frame is non-redundant, i.e., it is a Riesz basis. If $( a b ) ^ { - 1 } > 1$ and $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame, then that frame is redundant, i.e., the representation (a1) is not unique. If $( a b ) ^ { - 1 } < 1$, then for any $g \in L ^ { 2 } ( \mathbf{R} )$, the collection $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ is incomplete. See [a9], [a10].

The time-frequency atom $g_{n,m}$ is said to be localized at time $na$ and frequency $m b$ since the Fourier transform of $g_{n,m}$ is given by $e ^ { 2 \pi i m n a b } e ^ { 2 \pi i m b x }\hat{ g} ( \gamma - m b )$. A window function $g$ is said to have "good localization" in time and frequency if both $g$ and its Fourier transform $\hat{g}$ decay rapidly at infinity. Good localization can be measured in various ways. One way is to require that $\| tg ( t ) \| _ { 2 } \| \gamma \hat{g} ( \gamma ) \| _ { 2 } < \infty$. This is related to the classical uncertainty principle inequality, which asserts that any function $g \in L ^ { 2 } ( \mathbf{R} )$ satisfies $\| \operatorname { tg } ( t ) \| _ { 2 } \| \gamma \hat{g} ( \gamma ) \| _ { 2 } \geq ( 4 \pi ) ^ { - 1 } \| g \| _ { 2 } ^ { 2 }$.

The Balian–Low theorem asserts that if $( a b ) ^ { - 1 } = 1$ and if $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame for $L ^ { 2 } ( \mathbf{R} )$, then $g$ cannot have good localization. Specifically: If $( a b ) ^ { - 1 } = 1$ and if $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms a frame for $L ^ { 2 } ( \mathbf{R} )$, then $\| t g ( t ) \| _ { 2 } \| \gamma \hat{g} ( \gamma ) \| _ { 2 } = \infty$, i.e., $g$ 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 $\{ g _ { n , m} : n , m \in \mathbf{Z} \}$ forms an orthonormal basis (cf. also Orthonormal system) for $L ^ { 2 } ( \mathbf{R} )$, 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