# Generalized wavelets and the generalized wavelet transform on $\mathbb{R}^d$ for the Heckman-Opdam theory

## Main Article Content

## Abstract

## Article Details

## References

[1] L.C.Andrews, Special functions of Mathematics for engineers, second edition, Oxford University Press-Oxford-Tokyo-Melbourn, 1998. Google Scholar

[2] W.R.Bloom and H.Heyer, Harmonic analysis of probability measures on hyper-groups, Walter de Grayter, Berlin, New-York 1995. Google Scholar

[3] I.Cherednik, Inverse Harish-Chandra transform and difference operators, Internat. Math. Res. Notices 15 (1997), 733–750. Google Scholar

[4] L.Gallardo and K.Trim`eche, Positivity of the Jacobi-Cherednik intertwining operator and its dual, Adv. Pure Appl. Math. 1 (2012), 163–194. Google Scholar

[5] P.Goupilland, A.Grossmann and J.Morlet, Cycle octave and related transforms in seismic signal analysis, Geoexploration 23 (1984-1985), 85–102. Google Scholar

[6] A.Graussmann and J.Morlet, Decomposition of Hardy functions into square inte- grable wavelets of constant shape, Soc. Int. Am. Math. (SIAM), J. Math. Analys. 15 (1984), 723–736. Google Scholar

[7] A.Hassini and K.Trim`eche, Wavelets and generalized windowed transforms associated with the Dunkl-Bessel-Laplace operator on Rd ×R+, Mediterr. J. Math. 12 (2015), 1323–1344. Google Scholar

[8] G.J.Heckman and E.M.Opdam, Root systems and hypergeometric functions, I. Compositio Math. 64 (1987), 329–352. Google Scholar

[9] A.Jouini and K.Trim`eche, Two versions of wavelets and applications, Narosa Publishing House, Pvt.Ltd, 2006. Google Scholar

[10] T.H.Koornwinder, A new proof of the Paley-Wiener type theorem for the Jacobi transform, Arkiv For Math. 13 (1) (1975), 145–159. Google Scholar

[11] 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), p. 27-48. Google Scholar

[12] E.M.Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121. Google Scholar

[13] B.Schapira, Contribution to the hypergeometric function theory of Heckman and Opdam; sharp estimates, Schwartz spaces, heat kernel, Geom. Funct. Anal. 18 (2008), 222–250. Google Scholar

[14] K.Trim`eche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, 1997. Google Scholar

[15] K.Trim`eche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integ. Transf. and Spec. Funct. 13 (2002), 17–38. Google Scholar

[16] K.Trim`eche, The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operator and the Heckman Opdam theory, Adv. Pure Appl. Math. 1 (2010), 293–323. Google Scholar

[17] K.Trim`eche, Harmonic analysis associated with the Cherednik operators and the Heckman-Opdam theory, Adv. Pure Appl. Math. 2 (2011), 23–46. Google Scholar

[18] K.Trim`eche, The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory attached to the root systems of type B2 and C2, Korean J. Math. 22 (4) (2014), 711–728. Google Scholar

[19] K.Trim`eche, Positivity of the transmutation operators and absolute continuity of their representing measures for a root system on Rd, Int. J. App. Math. 28 (4) (2015), 427–453. Google Scholar

[20] K.Trim`eche, The harmonic analysis associated to the Heckman-Opdam theory and its application to a root system of type BCd., Preprint. Faculty of Sciences of Tunis. 2015. Google Scholar