There are several analogs of Cauchy’s integral transform in the theory of functions of several complex variables. Fantappie’s transform is one of them, possibly the simplest and oldest, with some remarkable properties we will discuss in detail. First, it is immediately connected to Radon transform, much exploited today in inverse problems. The characterization of Fantappie transforms of positive measures resonates with Bernstein’s theorem linking Laplace transforms and completely monotonic functions. One of the spectacular applications of Fantappie’s transform is the multivariate analog of Koethe–Grothendieck duality of spaces of analytic functions. The multiplicative structure of Fantappie’s transform brings us to the classical Markov–Krein correspondence, much investigated these days by probabilists.

An application to mathematical economics will be sketched.

The video of this talk is available on the IISc Math Department channel.