A decomposition of the Schwartz class by a derivative space and its complementary space

Abstract

Let $\mathcal{D} (R^n)$ be the class of all $C^{\infty}$–functions on $R^n$ with compact support. For a multi-index $\alpha$ we denote $\mathcal{D}^{\alpha} (R^n) = \{\mathcal{D}^{\alpha} \varphi : \varphi \in \mathcal{D} (R^n)\}$. We give a direct sum decomposition of $\mathcal{D} (R^n)$ by $\mathcal{D}^{\alpha} (R^n)$ and its complementary space.