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

DOI:
https://doi.org/10.1090/S0002-9939-00-05689-6

Published electronically:
October 20, 2000

MathSciNet review:
1814151

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a generalized frame in a separable Hilbert space indexed by a measure space , and assume its analysing operator is surjective. It is shown that is essentially discrete; that is, the corresponding index measure space can be decomposed into atoms such that is isometrically isomorphic to the weighted space of all sequences of complex numbers with , where This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in .

**[1]**Aliprantis, C. D., and Shaw, O. B., Principles of Real Analysis, 2nd Edition, Academic Press, New York 1990. MR**91c:28001****[2]**Chui, C. K., An Introduction to Wavelets, Academic Press, New York, 1992. MR**93f:42055****[3]**Daubechies, I., Ten Lectures on Wavelets, Soc. Ind. Appl. Math. Philadelphia, Pennsylvania 1992. MR**93e:42045****[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]**Heil, C. E., and Walnut, D. F., Continuous and discrete wavelet transforms, SIAM Review, 31(4) (1989), 628-666. MR**91c:42032****[7]**Hernandez, E., and Weiss, G. A., A First Course on Wavelets, CRC Press, 1996. MR**97i:42015****[8]**Kaiser, G., Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis, North-Holland, Amesterdam, 1990 a. MR**92a:81002****[9]**Kaiser, G., Generalized wavelet transforms, part I: The window -ray transforms, Technical Reports Series 18, Mathematics Department, University of Lowell. Part II: The multivariate analytic-signal transform, Technical Reports series 19, Mathematics Department, University of Lowell, 1990 b.**[10]**Kaiser, G., A Friendly Guide to Wavelets, Birkhäuser, second printing, 1995. MR**95i:94003****[11]**Mallat, S., A Wavelet Tour of Signal Processing, Academic Press, San Diego 1998. MR**99m:94012****[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]**Rioul, O., and Duhamel, P., Fast algorithms for discrete and continuous wavelet transforms, IEEE Transactions on Information Theory, 38 (2) (1992), 569-586. MR**93b:65215****[16]**Vetterli, M., and Kovacevi , J., Wavelets and Subband Coding, Prentice-Hall, Englewood Cliffs, 1995.**[17]**Yosida, K., Functional Analysis, Sixth Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR**82i:46002****[18]**Young, R. M., An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980. MR**81m:42027**

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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