Ranges and Inversion Formulas for Spherical Mean Operator and Its Dual

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

“…This problem was taken forward by the authors in [4] for the Riemann-Liouville operator and its dual. 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]).…”

Sonine Transform Associated to the Dunkl Kernel on the Real Line

“…Next, we establish for them the same results as those given in [8,14] for the Radon transform and its dual; and in [9] for the spherical mean operator and its dual on R. Especially:…”

“…-We give inversion formulas for S α,β and t S α,β associated with integro-differential and integro-differential-difference operators when applied to some Lizorkin spaces of functions (see [9,1,13]). …”

Spaces of DLp type and a convolution product associated with the spherical mean operator

“…For more details see [3,6,10,11]. We denote by (A) Ᏹ * (R × R n ) the space of infinitely differentiable functions on R × R n , even with respect to the first variable, (B) S n the unit sphere in R × R n , S n = (η,ξ) ∈ R × R n ; η 2 + ξ 2 = 1 , (2.1)…”

“…This operator plays an important role and has many applications, for example, in image processing of so-called synthetic aperture radar (SAR) data (see [7,8]), or in the linearized inverse scattering problem in acoustics [6]. In [10], the authors associate to the operator a Fourier transform and a convolution product and have established many results of harmonic analysis (inversion formula, Paley-Wiener and Plancherel theorems, etc. ).…”