**ABSTRACT**
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Bracket Products for Weyl-Heisenberg Frames

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P. Casazza

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Mathematics Department, 202 Mathematical Sciences Building, University of Missouri,
Columbia MO, 65211 USA

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M. Lammers

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Department of Mathematics, Western Washington University, Bond Hall 216, Bellingham, WA 98225,
USA

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.