Radon's inversion formulas

Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function f of two real variables ( formulas (III) and (III') in J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.- Nat. kl. 69 ( 1917), 262- 277) under the assumption that f is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that f be in L-p(R-2) for some p satisfying 4/3 < p < 2 suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon's inversion formulas.