Letter to the Editor
- Comment on “Continuous g-Frame in Hilbert -Modules”, Zhong-Qi Xiang
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 243453, 2 pages
Published 8 May 2013
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 361595, 20 pages
http://dx.doi.org/10.1155/2011/361595
Research Article
Continuous g-Frame in Hilbert -Modules
1Department of Mathematics, Science and Research Branch, Islamic Azad University, Kerman 7635131167, Iran
2Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 7616914111, Iran
Received 18 August 2010; Revised 23 November 2010; Accepted 19 January 2011
Academic Editor: H. B. Thompson
Copyright © 2011 Mehdi Rashidi Kouchi and Akbar Nazari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Transactions of the American Mathematical Society, vol. 72, pp. 341–366, 1952. View at Google Scholar · View at Zentralblatt MATH
- I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” Journal of Mathematical Physics, vol. 27, no. 5, pp. 1271–1283, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- V. K. Goyal, M. Vetterli, and N. T. Thao, “Quantized overcomplete expansions in analysis, synthesis, and algorithms,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 16–31, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992.
- J. J. Benedetto and D. Colella, “Wavelet analysis of spectrogram seizure chirps,” in Wavelet Applications in Signal and Image Processing III, vol. 2569 of Proceedings of SPIE, pp. 512–521, San Diego, Calif, USA, 1995.
- J. J. Benedetto and G. E. Pfander, “Wavelet periodicity detection algorithms,” in Wavelet Applications in Signal and Image Processing VI, vol. 3458 of Proceedings of SPIE, pp. 48–55, San Diego, Calif, USA, 1998. View at Publisher · View at Google Scholar
- P. G. Casazza, “The art of frame theory,” Taiwanese Journal of Mathematics, vol. 4, no. 2, pp. 129–201, 2000. View at Google Scholar · View at Zentralblatt MATH
- P. J. S. G. Ferreira, “Mathematics for multimedia signal processing, II: Discrete finite frames and signal reconstruction,” in Signal Processing for Multimedia, J. S. Byrnes, Ed., pp. 35–54, IOS Press, 1999. View at Google Scholar
- L. R. Neira and A. G. Constantinides, “Power spectrum estimation from values of noisy autocorrelations,” Signal Processing, vol. 50, no. 3, pp. 223–231, 1996. View at Publisher · View at Google Scholar
- J. J. Benedetto and W. Heller, “Irregular sampling and the theory of frames. I,” Note di Matematica, vol. 10, supplement 1, pp. 103–125, 1990. View at Google Scholar · View at Zentralblatt MATH
- H. G. Feichtinger and K. Gröchenig, “Theory and practice of irregular sampling,” in Wavelets: Mathematics and Applications, Stud. Adv. Math., pp. 305–363, CRC, Boca Raton, Fla, USA, 1994. View at Google Scholar · View at Zentralblatt MATH
- O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2003.
- N. F. Dudley Ward and J. R. Partington, “A construction of rational wavelets and frames in Hardy-Sobolev spaces with applications to system modeling,” SIAM Journal on Control and Optimization, vol. 36, no. 2, pp. 654–679, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- E. J. Candès, “Harmonic analysis of neural networks,” Applied and Computational Harmonic Analysis, vol. 6, no. 2, pp. 197–218, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. C. Eldar and G. D. Forney Jr., “Optimal tight frames and quantum measurement,” IEEE Transactions on Information Theory, vol. 48, no. 3, pp. 599–610, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- R. H. Chan, S. D. Riemenschneider, L. Shen, and Z. Shen, “Tight frame: an efficient way for high-resolution image reconstruction,” Applied and Computational Harmonic Analysis, vol. 17, no. 1, pp. 91–115, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P. G. Casazza and J. Kovačević, “Equal-norm tight frames with erasures,” Advances in Computational Mathematics, vol. 18, no. 2–4, pp. 387–430, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- V. K. Goyal, J. Kovačević, and J. A. Kelner, “Quantized frame expansions with erasures,” Applied and Computational Harmonic Analysis, vol. 10, no. 3, pp. 203–233, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. C. Lozano, J. Kovacevic, and M. Andrews, “Quantized frame expansions in a wireless environment,” in Proceedings of the Data Compression Conference (DCC '02), pp. 480–489, Snowbird, Utah, USA, March 2002.
- R. B. Holmes and V. I. Paulsen, “Optimal frames for erasures,” Linear Algebra and Its Applications, vol. 377, pp. 31–51, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- T. Strohmer and R. W. Heath Jr., “Grassmannian frames with applications to coding and communication,” Applied and Computational Harmonic Analysis, vol. 14, no. 3, pp. 257–275, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P. G. Casazza and J. Kovacević, “Uniform tight frames for signal processing and communication,” in Wavelets: Applications in Signal and Image Processing IX, vol. 4478 of Proceedings of SPIE, pp. 129–134, Diego, Calif, USA, July 2001. View at Publisher · View at Google Scholar
- G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser, Boston, Mass, USA, 1994.
- W. Sun, “-frames and -Riesz bases,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 437–452, 2006. View at Publisher · View at Google Scholar
- M. Frank and D. R. Larson, “Frames in Hilbert -modules and -algebras,” Journal of Operator Theory, vol. 48, no. 2, pp. 273–314, 2002. View at Google Scholar
- A. Khosravi and B. Khosravi, “Frames and bases in tensor products of Hilbert spaces and Hilbert -modules,” Proceedings of the Indian Academy of Sciences Mathematical Sciences, vol. 117, no. 1, pp. 1–12, 2007. View at Publisher · View at Google Scholar
- A. Khosravi and B. Khosravi, “Fusion frames and -frames in Hilbert -modules,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 6, no. 3, pp. 433–446, 2008. View at Publisher · View at Google Scholar
- E. C. Lance, Hilbert C∗-Modules: A Toolkit for Operator Algebraist, vol. 210 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1995. View at Publisher · View at Google Scholar
- I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-Trace C∗-Algebras, vol. 60 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1998.
- X.-C. Xiao and X.-M. Zeng, “Some properties of -frames in Hilbert -modules,” Journal of Mathematical Analysis and Applications, vol. 363, no. 2, pp. 399–408, 2010. View at Publisher · View at Google Scholar
- S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, “Continuous frames in Hilbert space,” Annals of Physics, vol. 222, no. 1, pp. 1–37, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, vol. 7 of Pure and Applied Mathematics, Interscience, New York, NY, USA, 1958.
- K. Yosida, Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 6th edition, 1980.
- L. Arambašić, “On frames for countably generated Hilbert -modules,” Proceedings of the American Mathematical Society, vol. 135, no. 2, pp. 469–478, 2007. View at Publisher · View at Google Scholar
- P. Găvruţa, “On the duality of fusion frames,” Journal of Mathematical Analysis and Applications, vol. 333, no. 2, pp. 871–879, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P. Găvruţa, “On some identities and inequalities for frames in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 469–478, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- R. Balan, P. Casazza, and D. Edidin, “On signal reconstruction without phase,” Applied and Computational Harmonic Analysis, vol. 20, no. 3, pp. 345–356, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- R. Balan, P. G. Casazza, D. Edidin, and G. Kutyniok, “A new identity for Parseval frames,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1007–1015, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- X. Zhu and G. Wu, “A note on some equalities for frames in Hilbert spaces,” Applied Mathematics Letters, vol. 23, no. 7, pp. 788–790, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH