The $q$-Dunkl wavelet packets

Yıl 2018, Cilt: 6 Sayı: 2, 311 - 320, 15.10.2018

Slim Bouaziz Kamel Mezlini Ahmed Fitouhi

Öz

Using the $q$-harmonic analysis associated with the $q$-Dunkl operator, we study three types
of $q$-wavelet packets and their corresponding $q$-wavelet transforms. We give for these wavelet transforms
the related Plancherel and inversion formulas as well as their $q$-scale discrete scaling functions.

Anahtar Kelimeler

q-Harmonic analysis, Packets

Kaynakça

• [1] N. Bettaibi and R. H. Bettaieb, q-Analogue of the Dunkl transform on the real line, Tamsui Oxford Journal of Mathematical Sciences, 25(2)(2007), 117-205
• [2] N. Bettaibi, R. H. Bettaieb and S. Bouaziz, Wavelet transform associated with the q-Dunkl operator, Tamsui Oxford Journal of Mathematical Sciences, 26(1) (2010) 77-101.
• [3] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, V. 160, SIAM, Philadelfia, PA, 1992.
• [4] A. Fitouhi, N. Bettaibi, Wavelet Transform in Quantum Calculus. J. Non. Math. Phys. 13, (2006), 492-506.
• [5] A. Fitouhi and R. H. Bettaieb, Wavelet Transform in the q2-Analogue Fourier Analysis, Math. Sci. Res. J. 12 (2008), no. 9, 202–214.
• [6] Grossman A and Morlet J, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723–736.
• [7] F. H. Jackson, On a q-Definite Integrals. Quarterly Journal of Pure and Applied Mathematics 41, 1910, 193-203.
• [8] T. H. Koornwinder, The continuous Wavelet Transform, Series in Approximations and decompositions, Vol. 1, Wavelets: An Elementary Treatment of Theory and Applications. Edited by T. H. Koornwinder, World Scientific, 1993; 27􀀀48.
• [9] T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333, 1992, 445-461.
• [10] R. L. Rubin, A q2􀀀Analogue Operator for q2􀀀 analogue Fourier Analysis, J. Math. Analys. App. 212; 1997;571􀀀582:
• [11] R. L. Rubin, Duhamel Solutions of non-Homogenous q2􀀀 Analogue Wave Equations, Proc. of Amer. Maths. Soc. V135; Nr 3; 2007; 777􀀀785:
• [12] F. Soltani, Fock spaces for the q-Dunkl kernel, The Advances in Pure Mathematics (APM), 2(3) (2012) pp. 169-176 DOI: 10.4236/apm.2012.23023
• [13] K. Trim`eche, Generalized harmonic analysis and wavelet packets, Gordon and Breach Science Publishers, 2001.
• [14] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
• [15] Hacısalihog˘lu, H. H., Diferensiyel geometri, Cilt I-II, Ankara U¨ niversitesi, Fen Faku¨ltesi Yayınları, 2000.
• [16] A. G¨org¨ul¨u and A. C. C¸ ¨oken, The Euler theorem for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclideanspace En+1 1 , Journ. Inst. Math. and Comp. Sci. (Math. Series) Vol:6, No.2 (1993), 161-165.