Classical Wavelet Transforms over Finite Fields

Subject Areas : History and biography

A. Ghaani Farashahi ¹ (Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Austria)

Received: 2015-10-17 Accepted : 2016-01-31 Published : 2015-12-01

Keywords: Finite field, classical wavelet group, classical wavelet transforms, dilation operators,

Abstract :

This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of classical wavelet coefficients.

References:

[1] 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.

[2] 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.

[3] 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.

[4] G. Caire, R.L. Grossman, and H. Vincent Poor. Wavelet transforms associated with finite cyclic groups. IEEE Transaction On Information Theory 39 (4), (1993), 113-119.

[5] F. Fekri, R.M. Mersereau, and R.W. Schafer. Theory of wavelet transform over finite fields. Proceedings of International Conference on Acoustics, Speech, and Signal Processing., 3, (1999), 1213-1216.

[6] K. Flornes, A. Grossmann, M. Holschneider, and B. Torresani. Wavelets on discrete fields. Appl. Comput. Harmon. Anal., 1, (1994), 137-146.

[7] A. Ghaani Farashahi, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc., to appear 2016.

[8] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489 (2016), 75-92.

[9] A. Ghaani Farashahi, Wave packet transform over finite fields, Electronic Journal of Linear Algebra, Volume 30 (2015), 507-529.

[10] A. Ghaani Farashahi. Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra 2 (1), (2015), 11-24.

[11] A. Ghaani Farashahi. Cyclic wave packet transform on finite Abelian groups of prime order. Int. J. Wavelets Multiresolut. Inf. Process., 12 (6), (2014), 1450041 (14 pages).

[12] A. Ghaani Farashahi and 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.

[13] A. Ghaani Farashahi and R. Kamyabi-Gol. Gabor transform for a class of non-abelian groups., Bull. Belg. Math. Soc. Simon Stevin., 19 (4), (2012), 683-701.

[14] C. P. Johnston. On the pseudodilation representations of flornes, grossmann, holschneider, and torresani. J. Fourier Anal. Appl., 3 (4), (1997), 377-385.

[15] G.L. Mullen and D. Panario. Handbook of Finite Fields., Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.

[16] R.J. McEliece. Finite Fields for Computer Scientists and Engineers., The Springer International Series in Engineering and Computer Science, 1987.

[17] O. Pretzel. Error-Correcting Codes and Finite Fields., Oxford Applied Mathematics and Computing Science Series, 1996.

[18] D. Ramakrishnan and R.J. Valenza. Fourier Analysis on Number Fields., Springer-Verlag, New York, 1999.

[19] R. Reiter and J.D. Stegeman. Classical Harmonic Analysis., 2nd Ed, Oxford University Press, New York, 2000.

[20] H. Riesel. Prime Numbers and Computer Methods for Factorization., (second edition), Boston: Birkhauser, 1994.

[21] S. Sarkar and H. Vincent Poor. Cyclic wavelet transforms for arbitrary finite data lengths. Signal Processing 80 (2000), 2541-2552.

[22] G. Strang and T. Nguyen. Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, 1996.

[23] 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.

[24] A. Vourdas. Harmonic analysis on a Galois field and its subfields. J. Fourier Anal. Appl., 14 (1), (2008), 102-123.

[25] M.W. Wong. Discrete Fourier Analysis. Pseudo-differential Operators Theory and applications Vol. 5, Springer-Birkhauser, 2010.