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

AbstractWe consider the Radon transformRα, α≥0, on the Laguerre hypergroupK=[0,+∞[×R. We characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such thatRαis a bijection from this space onto itself. We establish an inversion formula and a Plancherel theorem for the operatorRα. Finally, by using the continuous wavelet transform on the Laguerre hypergroupK, we deduce another expression for the inverseRα−1of the operatorRα