Injectivity of rotation invariant windowed Radon transforms (Q819718)

The author considers rotation invariant windowed Radon transforms that integrate a function over hyperplanes by using a radial weight (called window). \textit{E. T. Quinto} [J. Math. Anal. Appl. 91, 510--522 (1983; Zbl 0517.44009)] proved an injectivity result for square integrable functions with compact support. However this can not be extended in general. In fact, when the Laplace transform of the window has a zero with positive real part \(\delta\) the windowed Radon transform is not injective on functions with a Gaussian decay at infinity, depending on \(\delta\). Here the author obtains conditions on the window that imply injectivity of the windowed Radon transform on functions with a more rapid decay than any Gaussian function.