A Generalized Radon Transform on the Plane

Abstract

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ_α, β, and the Jacobi polynomials $P_{k}^{(\beta,\,\alpha)}(t)$. The transform R _α, β and its dual $R_{\alpha,\,\beta}^{\ast}$ are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in $L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+})$ are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations $Lu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f$ with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ_α, β. Another application is that, by an invariant property of R _α, β, a new product formula for the Jacobi polynomials of the type $P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)}$ is obtained.