The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.