Fonctions intérieurs dans la boule unité de \({\mathbb C}^ n\) dont les fonctions traces sont aussi intérieurs. (Inner functions in the unit ball of \({\mathbb C}^ n\) where the trace functions are also inner functions) (Q1093047)

We solve a problem set by W. Rudin, concerning inner functions. We prove the existence of inner functions in the unit ball B of \({\mathbb{C}}^ n\), for which every slice function is inner too. Aleksandrov's original construction of an inner function involved a series which converge in the Hardy space \(H^{1/2}(B)\). We refine his method in that way that our construction involves a series of basic functions which converges in all spaces \(H^{1/2}(D)\) where D runs through the family of discs containing the origin. This requires a very precise knowledge of the localization of the sets of points defining our basic functions.