Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets

Abstract

We consider the Radon transform R , ␣ G 0, on the Laguerre hypergroup ␣ w w Ks 0, qϱ ‫.ޒ=‬ We characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that R is a bijection ␣ from this space onto itself. We establish an inversion formula and a Plancherel theorem for the operator R . Finally, by using the continuous wavelet transform

Riesz Transform on the Dual of the Laguerre Hypergroup

“…As basic references of the Laguerre hypergroup, we refer the reader to (M. Assal & H. Ben Abdallah, 2005; M. M. Nessibi & K. Trimeche, 1997).…”

“…We recall for (λ, m) ∈K and for a suitable function f : K −→ C, the Fourier-Laguerre transform is defined in (M. M. Nessibi & K. Trimeche, 1997;M. Assal & H. Ben Abdallah, 2005) by:…”

“…It has been proved in M. M. Nessibi & K. Trimeche (1997) that the Fourier-Laguerre transform given above is a topological isomorphism from S (K) to S (K) and its inverse is given by…”

Bessel potential space on the Laguerre hypergroup

“…In this section we set some notations and we recall some basic results in harmonic analysis related to Laguerre hypergroups (see [4][5][6]). …”

Inversion of the Riemann-Liouville operator and its dual using wavelets

“…Indeed, they have proved the same results given by Ludwig, Helgason and Solmon for the classical Radon transform on R 2 [15,21,26] and for the spherical mean operator in [23], more precisely they have established that the Riemann-Liouville operator and its dual are isomorphisms on some subspaces of S e (R 2 ) and they have provided their inversion formulas in terms of integro-differential operators. Herein, we invert R α and t R α using generalized wavelets associated to the Riemann-Liouville operator and classical wavelets (see [24,29]). These new expressions are advantageous because of the large choice of wavelets, that are recognized as a powerful new mathematical tool in many areas, for example signal and image processing, time series analysis, geophysics ( [8,[11][12][13]).…”