Osculatory Interpolation in $\mathbb{R}^n $

In this work we study the Hermite osculatory interpolation problems in several variables. We give general Lagrange formulae for the solution of these problems and exhibit examples where these formulae apply in the context of $P_m $-correct Hermite problems. We finally give an inductive characterization of a family of $P_m $-correct Hermite problems, which we call totally reducible.