2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219102Let $G$ be a compact subgroup of $GL_n(\R)$ acting linearly on a finite dimensional vector space $E$. B. Malgrange has shown that the space $\mathcal{C}^\infty(\R^n,E)^G$ of $\mathcal{C}^\infty$ and $G$-covariant functions is a finite module over the ring $\mathcal{C}^\infty(\R^n)^G$ of $\mathcal{C}^\infty$ and $G$-invariant functions. First, we generalize this result for the Schwartz space $\mathscr{S}(\R^n,E)^G$ of $G$-covariant functions. Secondly, we prove that any $G$-covariant distribution can be decomposed into a sum of $G$-invariant distributions multiplied with a fixed family of $G$-covariant polynomials. This gives a generalization of an Oksak result proved in ([O]).Representation Theory46F05; 58C99; 58C81; 58C46.text