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 (ψ _R⁺ ,ψ _R̄ ) 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 (ψ _k̄ ,ψ _k ⁺ , k = {0,±1, ±2, ⋯}). The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space (S^?(ℝ)), the successive derivatives of the n-th power of f (n ∈ℤ and n ≠ 0) can be decomposed using only ψ _k ⁺ (Lemma); or if f in a subset of S^?(ℝ), called s?(ℝ), ψ _k⁺ and ψ _k̄ (k ∈ 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.