A generalized Radon transform on the plane
The authors define a new generalized Radon transform \(R_{\alpha,\beta}\) on the plane for functions even on each variable. Such \(R_{\alpha,\beta}\) has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator \(\Delta_{\alpha\beta}\), and the Jacobi polynomial \(P^{(\beta, \alpha)}_k(t)\). The authors obtain some properties of the transform \(R_{\alpha,\beta}\), as well as its dual operator \(R^*_{\alpha,\beta}\). Particularly, the authors obtain the generalized Fuglede formula and some inverse formulas for \(R_{\alpha,\beta}\) on the function space \(L^p_{\alpha,\beta}(\mathbb{R}^2_+)\) in terms of the bivariate Hankel-Riesz potentials. Moreover, the transform \(R_{\alpha,\beta}\) is used to represent the solutions of the partial differential equations \[ \sum^m_{j=1} a_j\Delta^j_{\alpha,\beta} u= f, \] and the Cauchy problem for the generalized wave equation associated with the operator \(\Delta_{\alpha,\beta}\).
scientific article; zbMATH DE number 5851416
scientific article