3. From Radon to Leray
Simon Gindikin · 2019
We discuss the influence of the Radon transform on multidimensional complex analysis. In the first turn, it is Leray’s construction of an universal multidimensional analogue of the Cauchy formula the Cauchy-Fantappie formula. It was extended by fundamental conceptions of analytic functionals, analytic duality, cohomology, and different complex versions of the Radon transform. In our exposition, the conception of the Cauchy-Radon transform plays an essential role. It is the result of the replacement of -function in the integrand of the Radon transform by a Cauchy kernel. It appears as a modification of the Radon transform for which the inversion formula is independent of the evenness of the dimension. This then gives new possibilities to consider analogs of the Radon transform in non-Euclidean situation.