UNIQUENESS AND NON-UNIQUENESS OF THE RADON TRANSFORM

. Let Rh = 0 , R is the Radon transform of h , Rh = ∫ R 2 δ ( p − α · x ) h ( x ) dx = ∫ L αp h ( s ) ds , where L αp is the straight line α · x = p , α = (cos θ, sin θ ) , 0 ≤ θ < 2 π . Uniqueness of R means that equation Rh = 0 implies h = 0 . Non-uniqueness means that there exists h , not equal to zero identically and satisfying equation Rh = 0 for all unit vectors α and all p ≥ 0 . We prove uniqueness of the Radon transform for h ∈ S ′ , where S ′ is the Schwartz’s space of tempered distributions. It is known that there are entire functions h , not equal to zero identically, h = h ( z ) , z = x 1 + ix 2 , which satisfy equation Rh = 0 .