Generalized frames and their redundancy
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- by A. Askari-Hemmat, M. A. Dehghan and M. Radjabalipour PDF
- Proc. Amer. Math. Soc. 129 (2001), 1143-1147 Request permission
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 $||\{c_i\}||^2=\sum |c_i|^2 w_i<\infty$, 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 $L^2(\mathbb {R})$.References
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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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1143-1147
- MSC (1991): Primary 42C15, 46C99
- DOI: https://doi.org/10.1090/S0002-9939-00-05689-6
- MathSciNet review: 1814151