Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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GENERALIZED WAVELETS AND THE GENERALIZED WAVELET TRANSFORM ON ℝd FOR THE HECKMAN-OPDAM THEORY

  • Hassini, Amina (Department of Mathematics Faculty of Sciences of Tunis University of El Manar CAMPUS) ;
  • Maalaoui, Rayaane (Department of Mathematics Faculty of Sciences of Tunis University of El Manar CAMPUS) ;
  • Trimeche, Khalifa (Department of Mathematics Faculty of Sciences of Tunis University of El Manar CAMPUS)
  • Received : 2016.01.13
  • Accepted : 2016.06.15
  • Published : 2016.06.30
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Abstract

By using the Heckman-Opdam theory on ${\mathbb{R}}^d$ given in [20], we define and study in this paper, the generalized wavelets on ${\mathbb{R}}^d$ and the generalized wavelet transform on ${\mathbb{R}}^d$, and we establish their properties. Next, we prove for the generalized wavelet transform Plancherel and inversion formulas.

Keywords

References

  1. L.C.Andrews, Special functions of Mathematics for engineers, second edition, Oxford University Press-Oxford-Tokyo-Melbourn, 1998.
  2. W.R.Bloom and H.Heyer, Harmonic analysis of probability measures on hypergroups, Walter de Grayter, Berlin, New-York 1995.
  3. I.Cherednik, Inverse Harish-Chandra transform and difference operators, Internat. Math. Res. Notices 15 (1997), 733-750.
  4. L.Gallardo and K.Trimeche, Positivity of the Jacobi-Cherednik intertwining operator and its dual, Adv. Pure Appl. Math. 1 (2012), 163-194.
  5. P.Goupilland, A.Grossmann and J.Morlet, Cycle octave and related transforms in seismic signal analysis, Geoexploration 23 (1984-1985), 85-102.
  6. A.Graussmann and J.Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, Soc. Int. Am. Math. (SIAM), J. Math. Analys. 15 (1984), 723-736.
  7. A.Hassini and K.Trimeche, Wavelets and generalized windowed transforms associated with the Dunkl-Bessel-Laplace operator on ${\mathbb{R}}^d{\time}{\mathbb{R}}_+$, Mediterr. J. Math. 12 (2015), 1323-1344. https://doi.org/10.1007/s00009-015-0540-4
  8. G.J.Heckman and E.M.Opdam, Root systems and hypergeometric functions, I. Compositio Math. 64 (1987), 329-352.
  9. A.Jouini and K.Trimeche, Two versions of wavelets and applications, Narosa Publishing House, Pvt.Ltd, 2006.
  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. https://doi.org/10.1007/BF02386203
  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.
  12. E.M.Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121. https://doi.org/10.1007/BF02392487
  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. https://doi.org/10.1007/s00039-008-0658-7
  14. K.Trimeche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, 1997.
  15. K.Trimeche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integ. Transf. and Spec. Funct. 13 (2002), 17-38. https://doi.org/10.1080/10652460212888
  16. K.Trimeche, 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.
  17. K.Trimeche, Harmonic analysis associated with the Cherednik operators and the Heckman-Opdam theory, Adv. Pure Appl. Math. 2 (2011), 23-46.
  18. K.Trimeche, The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory attached to the root systems of type $B_2$ and $C_2$, Korean J. Math. 22 (4) (2014), 711-728. https://doi.org/10.11568/kjm.2014.22.4.711
  19. K.Trimeche, Positivity of the transmutation operators and absolute continuity of their representing measures for a root system on ${\mathbb{R}}^d$, Int. J. App. Math. 28 (4) (2015), 427-453.
  20. K.Trimeche, The harmonic analysis associated to the Heckman-Opdam theory and its application to a root system of type $BC_d$., Preprint. Faculty of Sciences of Tunis. 2015.

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