Bracket Products for Weyl-Heisenberg Frames

P. Casazza

Mathematics Department, 202 Mathematical Sciences Building, University of Missouri, Columbia MO, 65211 USA
email: pete@math.missouri.edu

M. Lammers

Department of Mathematics, Western Washington University, Bond Hall 216, Bellingham, WA 98225, USA
email: lammerm@cc.wwu.edu

We provide a detailed development of the $L^1$ function-valued inner product on $L^2 (\mathbb R)$ known as the bracket product. In addition to some of the more basic properties, we show that this inner product has a Bessel's inequality, a Riesz Representation Theorem, and a Gram--Schmidt process. We then apply this to Weyl--Heisenberg frames to show that there exist "compressed" versions of the frame operator, the frame transform and the preframe operator. Finally, we introduce the notion of an $a$-frame and show that there is an equivalence between the frames of translates for this function-valued inner product and Weyl--Heisenberg frames.