Differentiation and the Balian{low Theorem
نویسندگان
چکیده
\If it is true, it can be proved." | Enrico Fermi 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 2imbt g(t ? na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then Z 1 ?1 jtg(t)j 2 dt Z 1 ?1 j ^ g()j 2 dd = +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) ^ () = 2ii ^ g(), the role of diierentiation 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 2ibm t g(t ? 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 2imbt g(t ? na)g that form exact frames, and a new proof of the BLT for exact frames that does not require diierentiation and relies only on classical real variable methods from harmonic analysis.
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