Classical wavelet systems over finite fields

Document Type: Research Paper

Author

University of Vienna

Abstract

This article presents an analytic approach to study admissibility conditions related to classical full wavelet systems over finite fields using tools from computational harmonic analysis and theoretical linear algebra. It is shown that for a large class of non-zero window signals (wavelets), the generated classical full wavelet systems constitute a frame whose canonical dual are classical full wavelet frames as well, and hence each vector defined over a finite field can be represented as a finite coherent sum of classical wavelet coefficients as well.

Keywords


[1] A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iran. Math. Soc.,
42(6)(2016), 1305 -1314.
[2] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35(3)(2013), 535-540.
[3] A. Arefijamaal and R.A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal., 19(3)(2009), 541-552.
[4] A. Arefijamaal and R.A. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process., 6(5) (2008), 749-759.
[5] A. Arefijamaal and R.A. Kamyabi-Gol, A Characterization of square integrable representations associated with CWT, J. Sci. Islam. Repub. Iran 18(2)(2007), 159-166.
[6] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis., IEEE Trans. Inform.
Theory, 36(5) (1990), 961-1005.
[7] K. Flornes, A. Grossmann, M. Holschneider, and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1(2)(1994), 137-146.
[8] A. Ghaani Farashahi, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc., to appear.
[9] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489(1) (2016), 75-92.
[10] A. Ghaani Farashahi, Classical wavelet transforms over finite fields, J. Linear Topol. Algebra, 4 (4) (2015), 241-257.
[11] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), 507-529.
[12] A. Ghaani Farashahi, Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra, 2 (1) (2015) 11-24.
[13] A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets
Multiresolut. Inf. Process., 12(6), 1450041 (14 pages), 2014.
[14] A. Ghaani Farashahi, M. Mohammad-Pour, A unified theoretical harmonic analysis approach to the cyclic
wavelet transform (CWT) for periodic signals of prime dimensions, Sahand Commun. Math. Anal., 1(2)(2014),
1-17.
[15] C. P. Johnston, On the pseudodilation representations of flornes, grossmann, holschneider, and torresani, J. Fourier Anal. Appl., 3(4)(1997), 377-385.
[16] G. L. Mullen, D. Panario, Handbook of Finite Fields, Series, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.
[17] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, The Springer International Series in Engineering and Computer Science, 1987.
[18] G. Pfander, Gabor Frames in Finite Dimensions, In Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
[19] O. Pretzel, Error-Correcting Codes and Finite Fields., Oxford Applied Mathematics and Computing Science
Series, 1996.
[20] R. Reiter and J.D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.
[21] H. Riesel, Prime numbers and computer methods for factorization, (second edition), Boston, Birkhauser, 1994.
[22] S. A. Vanstone and P. C. Van Oorschot, An Introduction to Error Correcting Codes with Applications, The Springer International Series in Engineering and Computer Science, 1989.
[23] A. Vourdas, Harmonic analysis on a Galois field and its subfields, J. Fourier Anal. Appl., 14(1)(2008), 102-123.