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The Fantappiè transform of function $f(x_1,x_2,\ldots,x_n)$, $x_1 \geqslant 0, \ldots, x_n\geqslant 0$, is defined by the formula $$ (\Phi f)(y) = \int\limits_{\mathbb R^n_+} \frac{f(x_1,\ldots,x_n)}{1+x_1 y_1 + \ldots + x_n y_n}dx_1\ldots dx_n, $$ where $\mathbb R^n_+ = \{ (x_1,\ldots,x_n) \in \mathbb R^n \mid x_1 \geqslant 0,\ldots,x_n \geqslant 0\}$. I know some practical applications of Laplace and Mellin transforms ─ two other integral transforms defined on functions on $\mathbb R^n_+$. I know also that modification of Fantappiè transform, the so called Cauchy-Fantappiè transform, is used for some purposes in complex analysis. But what are the practical applications of the Fantappiè transform? Was it used somewhere in mathematics besides complex analysis (I mean the Cauchy-Fantappiè transform)?

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Here a Fantappie transformation it used in investigating the moment problem - as well as proving some unexpected results in geometry of non-convex polyhedra:

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