BIORTHOGONAL SERIES EXPANSIONS OF THE X-RAY AND K-PLANE TRANSFORMS

Radon and x-ray transforms in R(r) serve as the mathematical model in several applications. The k-plane transform encompasses both transform ations. Within this general framework we consider biorthogonal series expansions in the domain and range of the k-plane transform with respe ct to some weighted function spaces for functions with support B-r, th e unit bail in R(r). By application of the well-known biorthogonal sys tem of Appell polynomials we prove that the determination of the expan sion coefficients leads to the solution of an infinite number of linea r equation systems. Furthermore, we are able to determine full explici t formulae for the k-plane transform of the basis functions. Finally, we give some estimates concerning the number of sampled directions whi ch are necessary to calculate approximations to a function from its k- plane data. The bounds for the number of directions are characterized using the dimensions of certain polynomial spaces.