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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 $L^2(\mathbb {R})$.
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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