Integral geometry in the sphere \(\mathbb{S}^d\) (Q2704084)

The author considers the Radon transform \(R\) on the sphere \(S^d\), \(d\geq 2\), defined by \textit{A. Abouelaz} and \textit{R. Daher} [Bull. Soc. Math. Fr. 121, No. 3, 353-382 (1993; Zbl 0798.44002)] and its dual \(R^*\). He characterizes the range of \(R^*\) and proves Paley-Wiener theorems on \(S^d\). Next he defines an integral transform \(R^{(\alpha, \beta)}\) which corresponds to a Radon transform on certain compact symmetric spaces of rank one. The dual transform \((R^{(\alpha, \beta)})^*\) and its main properties were given by \textit{T. Koornwinder} [Ark. Math. 13, 145-159 (1975; Zbl 0303.42022)], which permits also to define and study easily the transform \(R^{(\alpha, \beta)}\). In the last section of the paper he resolves the Poisson equation on \(S^d\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].