^{}Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna

Abstract

Finite affine groups are given by groups of translations and di- lations on ﬁnite cyclic groups. For cyclic groups of prime order we develop a time-scale (wavelet) analysis and show that for a large class of non-zero window signals/vectors, the generated full cyclic wavelet system constitutes a frame whose canonical dual is a cyclic wavelet frame.

[1] A. Arefijamaal, R. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal. 19 (2009), no. 3, 541-552. [2] A. Arefijamaal, R. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), no. 5, 749-759. [3] A. Arefijamaal, R. Kamyabi-Gol, A characterization of square integrable representations associated with CWT, J. Sci. Islam. Repub. Iran 18 (2007), no. 2, 159-166. [4] G. Caire, R. L. Grossman and H. Vincent Poor, Wavelet transforms associated with finite cyclic Groups, IEEE Trans. Information Theory, Vol. 39, No. 4, 1993. [5] P. Casazza and G. Kutyniok. Finite Frames Theory and Applications. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhauser. 2013. [6] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. [7] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992. [8] I. Daubechies and B. Han. The canonical dual frame of a wavelet frame. Appl. Comput. Harmon. Anal., 12(3):269-285, 2002. [9] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Sci. 72 (1952) 341-366. [10] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, Vol. 10 Academic Press, San Diego, 1999. [11] K. Flornes, A. Grossmann, M. Holschneider and B. Torresani, ´ Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1(1994),137-146. [12] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995. [13] S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis. Springer, 2013.

[14] A. Ghaani Farashahi, Wave packet transforms over finite fields, to appear, 2015. [15] 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 pp. [16] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups, Bull. Malays. Math. Sci. Soc., 38(2)(2015), 779-803. [17] A. Ghaani Farashahi, R. Kamyabi-Gol, Gabor transform for a class of non-abelian groups, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 4, 683-701. [18] 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. [19] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press, 1979. [20] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal., 4 (1997), 380-413. [21] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol 1, 1963. [22] C. P. Johnston, On the pseudodilation representations of flornes, grossmann, holschneider, and torr´esani, J. Fourier Anal. Appl., 3(4)(1997), 377-385. [23] G. L. Mullen and D. Panario, Handbook of Finite Fields, Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013. [24] G. Pfander. Gabor Frames in Finite Dimensions, In G. E. Pfander, P. G. Casazza, and G. Kutyniok, editors, Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013. [25] G. Pfander and H. Rauhut, Sparsity in time-frequency representations, J. Fourier Anal. Appl., 11(6)(2010), 715- 726. [26] G. Pfander, H. Rauhut, and J. Tropp, The restricted isometry property for time-frequency structured random matrices, Probability Theory and Related Fields, 3-4(156)(2013), 707-737. [27] G. Pfander, H. Rauhut, and J. Tanner, Identification of matrices having a sparse representation, IEEE Transactions on Signal Processing, 56 (11)(2008), 5376-5388. [28] H. Rauhut, Compressive sensing and structured random matrices, In M. Fornasier, editor, Theoretical foundations and numerical methods for sparse recovery, volume 9 of Radon Series Comp. Appl. Math., pp 1-92. deGruyter, 2010. [29] H. Riesel, Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhauser, ISBN 0-8176-3743-5, 1994. [30] S. Sarkar, H. Vincent Poor, Cyclic Wavelet Transforms for Arbitrary Finite Data Lengths, Signal Processing, 80 (2000), 2541-2552. [31] G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996. [32] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, ISBN 0-13-097080-8, 1995.