Abstract
We provide a detailed development of theL 1function-valued inner product onL 2(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.
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Casazza, P.G., Lammers, M.C. (2003). Bracket Products for Weyl—Heisenberg Frames. In: Feichtinger, H.G., Strohmer, T. (eds) Advances in Gabor Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0133-5_4
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DOI: https://doi.org/10.1007/978-1-4612-0133-5_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6627-3
Online ISBN: 978-1-4612-0133-5
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