# Advances in Analysis

### The *q*-Bessel Wavelet Packets

Download PDF (559.1 KB) PP. 27 - 39 Pub. Date: July 8, 2016

### Author(s)

**Slim Bouaziz**^{*}

Department of Mathematics, Preparatory Institute of Engineer Studies of El-Manar, Tunis, Tunisia

### Abstract

*q*-harmonic analysis associated with the

*q*-Bessel operator, we study some types of

*q*-wavelet packets and their corresponding q-wavelet transforms. We give for these wavelet transforms the related Plancheral and inversion formulas as well as their

*q*-scale discrete scaling functions.

### Keywords

*q*-Harmonic analysis, packets, wavelets.

### References

[1] G. E. Andrews, q-Series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, Regional Conf. Ser. in Math., no. 66, Amer. Math. Soc., Providence, R. I. 1986.

[2] G. E. Andrews and R. Askey, Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials, Higher combinatorics, M. Aigner, ed., Reidel, (1977), pp. 3-26.

[3] L. D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (4) (2004), 1197-1203.

[4] N. Bettaibi and R. H. Bettaieb, q-Aanalogue of the Dunkl transform on the real line, Tamsui Oxford Journal of Mathematical Sciences, 25(2)(2007), 117-205.

[5] N. Bettaibi, F. Bouzeffour, H. Ben Elmonser, W. Binous, Elements of harmonic analysis related to the third basic zero order Bessel function, J. Math. Anal. Appl., V 342, Issue 2, (2008), 1203-1219.

[6] N. Bettaibi, K. Mezlini and M. El Guénichi, On Rubin’s harmonic analysis and its related positive definite functions, Acta Mathematica Scientia, 32B(5), (2012) 1851-1874.

[7] H. Exton, A basic analogue of the Bessel-Clifford equation, Jnanabha 8, (1978), 49-56.

[8] H. Exton, q-Hypergeometric Functions and Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1983.

[9] H. Exton and H. M. Srivastava, A certain class of q-Bessel polynomials, Math. Comput. Modelling 19 (2) (1994), 55-60.

[10] A. Fitouhi, N. Bettaibi, Wavelet Transform in Quantum Calculus. J. Non. Math. Phys. 13, (2006), 492-506.

[11] A. Fitouhi, N. Bettaibi, W. Binous, Inversion formulas for the q-Riemann-Liouville and q-Weyl transforms using wavelets, Fractional Calculus and Applied Analysis, V 10, Nr 4, (2007).

[12] A. Fitouhi and R. H. Bettaieb, Wavelet Transform in the q 2-Analogue Fourier Analysis, Math. Sci. Res. J. 12 (2008), no. 9, 202U214.

[13] A. Fitouhi, M. M. Hamza, F. Bouzeffour, The q ? j Bessel Function, J. Approx. Theory, V 115, Issue 1, (2002), 144-166.

[14] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990).

[15] M. E. H. Ismail, The zeros of basic Bessel functions, the Function J+ax(x), and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1–19.

[16] M.E.H. Ismail, On Jackson?s Third q-Bessel Function and q-Exponentials, Preprint, 2001.

[17] W. Hahn, Die mechanische Deutung einer geometrischen Differenzengleichung, Zeitschrift für Angewandte Mathematik und Mechanik 33, (1953), 270-272.

[18] F. H. Jackson, On a q-Definite Integrals. Quarterly Journal of Pure and Applied Mathematics 41, 1910, 193-203.

[19] V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, (2002).

[20] T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333, 1992, 445-461.

[21] C. Krattenthaler and H. M. Srivastava, Summations for basic hypergeometric series involving a q-analogue of the Digamma function, Comput. Math. Appl. 32 (3) (1996), 73-91.

[22] R. L. Rubin, A q_{2}?Analogue Operator for q_{2}- analogue Fourier Analysis, J. Math. Analys. App.
212, 1997, 571-582.

[23] R. L. Rubin, Duhamel Solutions of non-Homogenous q_{2}- Analogue Wave Equations, Proc. of Amer. Maths.
Soc. V135, Nr 3, 2007, 777 - 785.

[24] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci. 5 (2011), 390-444.

[25] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Inc., First edition 2012.

[26] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.

[27] K. Trimèche, Generalized harmonic analysis and wavelet packets, Gordon and Breach Science Publishers, 2001.