ABSTRACT
In this chapter, the author adopt the geometric approach based upon the dual Radon transform on the sphere. In particular, they show the transmutation theorem (Theorem 1) for d ⩾ 5: this theorem is an improvement of [2, Theorem 2.2]. Then, they prove an improvement of the Paley-Wiener theorems; as the application they give an explicit solution for the Poisson equation. The authors characterize the range of the dual Radon transform. They study the Paley-Wiener theorems in the sphere and give some applications of Paley-Wiener theorems.
