1999/11/04 by Peter G. Casazza, Casazza, Peter G., M. C. Lammers +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #math.FA
paper · pdf · doi:10.48550/arxiv.math/9911026
37 pages
arxiv created 1999/11/04 · openalex publication_date 1999/11/04 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions (gn) into a sequence (en) with the property that (Emben)m,n∈ \Bbb Z is orthonormal in L2(\Bbb R). Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions g∈ L2(\Bbb R) and ab=1 so that the family (EmbTnag) is complete in L2(\Bbb R). One consequence of this is that for functions g supported on a half-line [α,∞) (in particular, for compactly supported g), (g,1,1) is complete if and only if sup0≤ t< a|g(t-n)|\not= 0 a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any g∈ L2(\Bbb R), A≤ ∑n |g(t-na)|2≤ B is equivalent to (Em/ag) being a Riesz basic sequence.