## Differentiation And The Balian-Low Theorem (1995)

Venue: | J. Fourier Anal. Appl |

Citations: | 34 - 18 self |

### BibTeX

@ARTICLE{Benedetto95differentiationand,

author = {John J. Benedetto and Christopher Heil and David F. Walnut},

title = {Differentiation And The Balian-Low Theorem},

journal = {J. Fourier Anal. Appl},

year = {1995},

volume = {1},

pages = {355--402}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The Balian--Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl g(fl)j 2 dfl ' = +1: The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g 0 ) (fl) = 2ßifl g(fl), the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form fe 2ßibm t g(t \Gamma an )g such that f(an ; bm )g has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems fe 2ßimbt g(t \Gamma na)g that form exact frames, and a new proof of the BLT for exact frame...