Characterizations of function spaces by the discrete Radon transform

Let ℤ ⁿ be the lattice in ℝ ⁿ and G the set of all discrete hyperplanes in ℤ ⁿ. Similarly, as in the Euclidean case, for a function f on ℤ ⁿ, the discrete Radon transform Rf is defined by the integral of f over discrete hyperplanes, and R maps functions on ℤ ⁿ to functions on G. In this paper, we determine the Radon transform images of the Schwartz space S(ℤ ⁿ), the space of compactly supported functions on ℤ , and a discrete Hardy space H ¹(ℤ ⁿ).