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.
decomposition energy operator schwartz space energy function n-th derivative