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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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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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  • [1] Charalambos D. Aliprantis and Owen Burkinshaw, Principles of real analysis, 2nd ed., Academic Press, Inc., Boston, MA, 1990. MR 1043168
  • [2] Charles K. Chui, An introduction to wavelets, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR 1150048
  • [3] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
  • [4] Duffin, R. J., and Schaeffer, A.C., A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. MR 13:839a
  • [5] Grossman, A., Kronland-Martinet, R., Morlet, J., Reading and understanding continuous wavelet transforms. In J. M. Combes, A. Grossmann, and Ph. Tchamitchian, editors, Wavelets, Time Frequency Methods and Phase Space, pages 2-20, Springer-Verlag, Berlin, 1989. Proceedings of International Colloquium on Wavelets and Applications, Marseille, France, Dec. 1987. CMP 21:16
  • [6] Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485,
  • [7] Eugenio Hernández and Guido Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR 1408902
  • [8] Gerald Kaiser, Quantum physics, relativity, and complex spacetime, North-Holland Mathematics Studies, vol. 163, North-Holland Publishing Co., Amsterdam, 1990. Towards a new synthesis. MR 1091248
  • [9] Kaiser, G., Generalized wavelet transforms, part I: The window $X$-ray transforms, Technical Reports Series $\char93 $ 18, Mathematics Department, University of Lowell. Part II: The multivariate analytic-signal transform, Technical Reports series $\char93 $ 19, Mathematics Department, University of Lowell, 1990 b.
  • [10] Gerald Kaiser, A friendly guide to wavelets, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1287849
  • [11] Stéphane Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, 1998. MR 1614527
  • [12] Pettis, B. J., On integration on vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
  • [13] Burrus, C. S., Gopinath, R. A., Guo, H., Introduction to Wavelet and Wavelet Transforms, Prentice Hall, New Jersey, 1998.
  • [14] Rioul, O., Fast computation of the continuous wavelet transform, Proc. Int. Conf. Acoust. Speech, Signal Processing, IEEE, Toronto, Canada, March 1991.
  • [15] Olivier Rioul and Pierre Duhamel, Fast algorithms for discrete and continuous wavelet transforms, IEEE Trans. Inform. Theory 38 (1992), no. 2, 569–586. MR 1162215,
  • [16] Vetterli, M., and Kovacevi $\acute{\mbox{c}}$, J., Wavelets and Subband Coding, Prentice-Hall, Englewood Cliffs, 1995.
  • [17] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
  • [18] Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684

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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

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