In the 6-volume "Generalized functions" a treatment of Fantappie transformations is promised in Vol. 1 (bottom of p.461 of the Russian edition) to come in Vol. 5. However, there is no Fantappie anywhere in Vol. 5.

A Fantappie transformation is an integral transform $$ \mu\mapsto\int_{\mathbb{R}^n} \frac{d\mu(X)}{(1-\langle T,X\rangle)^{d+1}} $$

Does anyone have an idea why the plan was changed?

My interest in this topic stems from not so well-known fact that certain Fantappie transforms of uniform measures supported on polyhedra are rational functions of $T$.