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Generalized frames and their redundancy


Authors: A. Askari-Hemmat, M. A. Dehghan and M. Radjabalipour
Journal: Proc. Amer. Math. Soc. 129 (2001), 1143-1147
MSC (1991): Primary 42C15, 46C99
Published electronically: October 20, 2000
MathSciNet review: 1814151
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Abstract:

Let $h$ be a generalized frame in a separable Hilbert space $H$ indexed by a measure space $(M,\mathcal{ S},\mu)$, and assume its analysing operator is surjective. It is shown that $h$ is essentially discrete; that is, the corresponding index measure space $(M,\mathcal{ S},\mu)$can be decomposed into atoms $E_1,E_2,\cdots$ such that $L^2(\mu)$ is isometrically isomorphic to the weighted space $\ell^2_w$ of all sequences $\{c_i\}$ of complex numbers with $\vert\vert\{c_i\}\vert\vert^2=\sum \vert c_i\vert^2 w_i<\infty$, where $w_i=\mu(E_i), i=1,2,\cdots.$ This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in $L^2(\mathbb{R})$.


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Additional Information

A. Askari-Hemmat
Affiliation: Department of Mathematics, University of Shiraz, Shiraz, Iran

M. A. Dehghan
Affiliation: Department of Mathematics, Valiasr University, Rafsanjan, Iran

M. Radjabalipour
Affiliation: Department of Mathematics, University of Kerman, Kerman, Iran
Email: radjab@arg3.uk.ac.ir

DOI: https://doi.org/10.1090/S0002-9939-00-05689-6
Keywords: Generalized frame, redundancy, wavelet, windowed Fourier transform
Received by editor(s): February 20, 1998
Received by editor(s) in revised form: October 12, 1998, and July 10, 1999
Published electronically: October 20, 2000
Additional Notes: This research is supported by Mahani Math. Research Center (Kerman, Iran) and ICTP (Trieste, Italy)
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society