On the determination of a function in the plane by its integrals over straight lines

Abstract

In the December 1957 issue of the American Mathematical Monthly (page 750) there appears the solution by D. J. Newman of the following problem proposed by him: Let f(x, y) be continuous and summable over the plane, and let the line integral ffds be zero when extended over any infinite straight line; prove that f 0. It was pointed out in the Monthly that this problem, which is related to the projections of probability distributions, had been stated and solved by A. Renyi [I]. In this note I shall give another short solution of this problem based on the use of the Newtonian potential, and show how the method applies to the solution of another problem raised by Renyi in the same paper. Let U(x, y, z) be the Newtonian potential of the distribution of charge in the x-y plane with charge density f(x, y), of which we momentarily assume if ?K. Let polar coordinates (r, 0) be introduced in the x-y plane at an arbitrary pole (x0, yo), and setf(x, y) =g(r, 0). Then