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.