Bracket products for WeylHeisenberg frames
Abstract
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 WeylHeisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a GramSchmidt process which turns a sequence of functions $(g_{n})$ into a sequence $(e_{n})$ with the property that $(E_{mb}e_{n})_{m,n\in \Bbb Z}$ is orthonormal in $L^{2}(\Bbb R)$. Armed with this inner product, we obtain several results concerning WeylHeisenberg 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 $g\in L^{2}(\Bbb R)$ and $ab=1$ so that the family $(E_{mb}T_{na}g)$ is complete in $L^{2}(\Bbb R)$. One consequence of this is that for functions $g$ supported on a halfline $[{\alpha},\infty)$ (in particular, for compactly supported $g$), $(g,1,1)$ is complete if and only if $\text{sup}_{0\le t< a}g(tn)\not= 0$ a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any $g\in L^{2}(\Bbb R)$, $A\le \sum_{n} g(tna)^{2}\le B$ is equivalent to $(E_{m/a}g)$ being a Riesz basic sequence.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1999
 arXiv:
 arXiv:math/9911026
 Bibcode:
 1999math.....11026C
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 37 pages