The Generalization of the Decomposition of Functions by Energy Operators

TL;DR

This paper extends the theory of energy operators by defining a family of differential energy operators and demonstrating their ability to uniquely decompose derivatives of functions in Schwartz space, with applications to energy functions.

Contribution

It introduces a generalized family of energy operators and proves their unique decomposition capability for derivatives of functions in Schwartz space.

Findings

Successive derivatives of function powers can be decomposed using energy operators.

Decomposition is unique under certain conditions.

Properties of the energy operators' kernel and image are characterized.

Abstract

This work starts with the introduction of a family of differential energy operators. Energy operators ($Psi_R^+$, $Psi_R^-$) 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 ($Psi_k^+$, $Psi_k^-$, k = {0,1,2,..}). The main part of the work is to demonstrate that for any smooth real-valued function f in the Schwartz space ($S^-(R)$), the successive derivatives of the n-th power of f (n in Z and n not equal to 0) can be decomposed using only $Psi_k^+$ (Lemma) or with $Psi_k^+$, $Psi_k^-$ (k in Z) (Theorem) in a unique way (with more restrictive conditions). 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.