The Generalization of the Decomposition of Functions by Energy Operators
J.-P.~Montillet
% \boldmath{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 ($\Psi^{-}_k$, $\Psi^{+}_k$, $k=\{0,\pm 1,\pm 2,...\}$). 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 $n\neq 0$) can be decomposed using only $\Psi^{+}_k$ (Lemma); or if $f$ in a subset of $\mathbf{S}^{-}(\mathbb{R})$, called $\mathbf{s}^{-}(\mathbb{R})$, $\Psi^{+}_k$ and $\Psi^{-}_k$ ($k\in \mathbb{Z}$) 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. %A corollary of the second Lemma ends the development of this paper. } %
Journal_IEEE2_add03.tex

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\begin{document}
%
%\begin{frontmatter}
%
\title{The Generalization of the Decomposition of Functions by Energy Operators}
% and the Application to the Linear Partial Differential Equations}
%
%\maketitle
% paper title
% can use linebreaks \\ within to get better formatting as desired
%\author{J.-P.~Montillet, jeanfi_montillet@yahoo.fr}
%GNSS Satellite Navigation Group, GPO Box U1987, Curtin University of Technology,
%\\Perth, Australia
%\\jeanfi_montillet@yahoo.fr
%\thanks{Dr J.-P.Montillet is currently with Curtin University of Technology, Department of GeoSpatial Sciences, GNSS Satellite Navigation Group, GPO Box U1987, Perth Western Australia, email: j.montillet@curtin.edu.au }% <-this % stops a space
%
\maketitle
%
\author{J.-P.~Montillet \footnote{Dr. J.P. Montillet is a Research Fellow at the Research School of Earth Sciences at the Australian National University}, $j.p.montillet@anu.edu.au$}%\author{J.-P.~Montillet, jeanfi_montillet@yahoo.fr}
%\title{Generalization of the Decomposition of Finite Energy Functions by Energy Operators and the Application to the Linear Partial Differential Equations}
%
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%
\begin{abstract}
%
\boldmath{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 ($\Psi^{-}_k$, $\Psi^{+}_k$, $k=\{0,\pm 1,\pm 2,...\}$). 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 $n\neq 0$) can be decomposed using only $\Psi^{+}_k$ (Lemma); or if $f$ in a subset of $\mathbf{S}^{-}(\mathbb{R})$, called $\mathbf{s}^{-}(\mathbb{R})$, $\Psi^{+}_k$ and $\Psi^{-}_k$  ($k\in \mathbb{Z}$) 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.
%A corollary of the second Lemma ends the development of this paper. 
}
%
\end{abstract}
% IEEEtran.cls defaults to using nonbold math in the Abstract.
% This preserves the distinction between vectors and scalars. However,
% if the journal you are submitting to favors bold math in the abstract,
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% of the abstract to achieve this. Many IEEE journals frown on math
% in the abstract anyway.

% Note that keywords are not normally used for peerreview papers.
%\begin{IEEEkeywords}
%Energy Operator, Teager-Kaiser, Energy of a function.
%\end{IEEEkeywords}

% For peer review papers, you can put extra information on the cover
% page as needed:
% \ifCLASSOPTIONpeerreview
% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
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%
%\section{Introduction}
%\IEEEPARstart{S}{ince} 2005 ....
%
%The Teager-Kaiser energy operator has been introduced in \cite{Kaiser90} and the past decade has seen numerous applications flourishing in signal processing \cite{Bovik93},\cite{Mitra et Al1991} and \cite{Lin95}; speech recognisation \cite{Kaiser et al.1993} and more exotic works such as mobile phone positioning \cite{Hamila1999} and mammals localization \cite{Kandia2006}. 
%
%The presented work is not based on the current main stream research work surrounding this operator. 
%
%\section{Main Conjecture}
%
\section{Introduction}\label{Introduction part}
%
Two decades ago, an energy operator (${\Psi}_{R}^{-}$) was first defined in \cite{Kaiser90}. Since then, this work has been extensively used in telecommunications (see for example \cite{Bovik93} or \cite{Hamila1999}). The bilinearity properties of this operator were studied in \cite{Boudraa et al.2009}. More recently, the author in \cite{JPMontillet2010} introduced the energy operators (${\Psi}_{R}^{+}$, ${\Psi}_{R}^{-}$) in time and space. This was part of a general method for separating the energy of finite energy functions in time and space with application to the wave equation. Note that calling ${\Psi}_{R}^{+}$ an energy operator is an abuse of language. The term was already used in \cite{JPMontillet2010} as the definition of ${\Psi}_{R}^{+}$ is very close to the Teager-Kaiser energy operator ${\Psi}_{R}^{-}$. 
%    
\newline  This work focuses on the decomposition of a smooth real-valued function $f$ using family of differentiable energy operators based on the energy operators ${\Psi}_{R}^{\pm}$. Throughout this work, $f$ is supposed to be in the Schwartz space $\mathbf{S}^{-}(\mathbb{R})$ defined as:
%
\begin{equation}
\mathbf{S}^{-}(\mathbb{R}) =\{f \in \mathbf{C}^{\infty}(\mathbb{R}), \qquad {sup}_{t<0} |t^k||\partial_t^j f(t)|<\infty,\qquad \forall k \in \mathbb{Z}^+, \qquad \forall j \in \mathbb{Z}^+ \}
\end{equation}
%
Note that $\partial_t$ is the derivative related to the variable $t$. $\mathbb{Z}^+$ is the set of positive integers including $0$. In the following, let us call the set $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$ all functions defined such as $F:$ $\mathbf{S}^{-}(\mathbb{R})$ $\rightarrow$ $\mathbf{S}^{-}(\mathbb{R})$.
%
%, and expandable in Taylor- Mc Laurin Series.
% Through the development, $f$ is supposed to be $n$ times differentiable and continue in $\mathbb{R}$ ($f \in C^{\infty}(f)$). 
% 
\newline First, two families of differential energy operators $\Psi^{+}_k$ and $\Psi^{-}_k$ ($k=\{0,\pm 1,\pm 2,...\}$) are introduced with the notations following \cite{Maragos1995} and \cite{JPMontillet2010} where $k$ is the degree of their derivatives. Note that for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, $\Psi^{+}_k(f)$ and $\Psi^{-}_k(f)$ ($k=\{0,\pm 1,\pm 2,...\}$) are in $\mathbf{S}^{-}(\mathbb{R})$ ($(\Psi^{\pm}_k)_{k \in \mathbb{Z} }$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$. If not explicitly written, any families of operator in $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$ in this work follow the derivative chain rule property:
%
%It is important to underline that the 
%
\begin{equation}\label{derivative chain rules0111}
\forall f \in \mathbf{S}^{-}(\mathbb{R}), \qquad \partial_t \Psi_k(f) = \Psi_{k+1}(f) + \Psi_{k-1}(\partial_t f)
\end{equation} 
%
In addition, the term \emph{decompose} is defined as:
%
%
\vspace{1.5em}
%
\newline $\bold{Definition}$ $1$: for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$, for all $v\in\mathbb{Z}^+-\{0\}$, for all  $n\in\mathbb{Z}^+$ and $n>1$, the family of operators $(\Psi_k)_{k \in \mathbb{Z}}$ (with $(\Psi_k)_{k \in \mathbb{Z}}$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$) decomposes $\partial_t^v$$f^n$ in $\mathbb{R}$, if it exists $(N_j)_{j\in \mathbb{Z}^+ \cup \{0\}}$ $\subseteq$ $\mathbb{Z^+}$,  $(C_i)_{i=-N_j}^{N_j}$ $\subseteq$ $\mathbb{R}$, and it exists $(\alpha_j)$ and $l$ in $\mathbb{Z^+}\cup\{0\}$ (with $l<v$) 
\\ such as $\partial_t^v$$f^n = \sum_{j=0}^{v-1} \big(_{j}^{v-1} \big) \partial_t^{v-1-j} f^{n-l} \sum_{k=-N_j}^{N_j} C_k \Psi_k(\partial_t^{\alpha_k}f)$.
%
%
%
\vspace{1.5em}
%
\\  Definition $1$ is based on the general Leibniz rule for the $n-th$ derivative of a product of functions \cite{BruceWest}. One can define the image $Im(\Psi^{+}_k)$ and kernel $Ker(\Psi^{+}_k)$ (for $k$ in $\mathbb{Z}$) of an energy operator such as:
%
\begin{equation}\label{Impsik+}
%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
Im(\Psi^{+}_k) = \{\Psi^{+}_k(f) \in \mathbb{R} | \qquad f \in \mathbf{S}^{-}(\mathbb{R}) \}
\end{equation}
%
%\\ Moreover, one can define the kernel of the energy operator  such that for $k$ in $\mathbb{Z}$:
and
%
\begin{equation}
%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
Ker(\Psi^{+}_k) = \{f\in \mathbf{S}^{-}(\mathbb{R})|  \qquad \Psi^{+}_k(f)=0 \}
\end{equation}
%
Obviously, the null function ($f: \mathbb{R}\rightarrow 0$) belongs to $Ker(\Psi^{+}_k)$.
% 
%
One can define also $Im(\Psi^{-}_k)$ and $Ker(\Psi^{-}_k)$ associated with the family of DEOs $\Psi^{-}_k$ ($k$ in $\mathbb{Z}$).
% In Section \ref{SectionProperties}, the study of the properties of the images helps to simplify the formulas shown in Lemma and  Theorem.
%
Now, let us define a subset $\mathbf{s}^{-}(\mathbb{R})$ $\subseteq$ $\mathbf{S}^{-}(\mathbb{R})$ such as:
%
\begin{eqnarray}
%
%
\mathbf{s}^{-}(\mathbb{R}) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})|  \hspace{0.2em}\forall\hspace{0.2em} k \in \mathbb{Z}, \hspace{0.2em}  \Psi^{+}_k(f) \neq \{0\} \hspace{0.2em}  \nonumber \\
                   & & \hspace{0.2em} \forall\hspace{0.2em} k \in \mathbb{Z}-\{1\}, \hspace{0.2em}\Psi^{-}_k(f) \neq \{0\} \} \nonumber  
%
\end{eqnarray}
%
Note that a possible way to define $\mathbf{s}^{-}(\mathbb{R})$ is:
%
\begin{equation}
\mathbf{s}^{-}(\mathbb{R}) = \{ f \in  \mathbf{S}^{-}(\mathbb{R}) | f \notin  (\cup_{k \in \mathbb{Z}} Ker(\Psi^{+}_k))\cup(\cup_{k \in \mathbb{Z}-\{1\}} Ker(\Psi^{-}_k))\}
\end{equation}
%
% to take out of  $\mathbf{S}^{-}(\mathbb{R})$ all $f$ in $Ker(\Psi^{-}_k)$ (for all $k$ in $\mathbb{Z}$$-$$\{1\}$) and $Ker(\Psi^{+}_p)$ (for all $p$ in $\mathbb{Z}$). 
%In a the definition of the subset $\mathbf{s}^{-}(\mathbb{R})$ excludes $\Psi^{-}_1$ as by definition of  this operator $Im(\Psi^{-}_1)$ equal $\{0\}$.
The definition of the subset $\mathbf{s}^{-}(\mathbb{R})$ excludes $\Psi^{-}_1$ as by definition of  this operator $Im(\Psi^{-}_1)$ equal $\{0\}$ for all $f$ in  $\mathbf{S}^{-}(\mathbb{R})$.
%
Following Definition $1$, the \emph{uniqueness} of the decomposition  in $\mathbf{s}^{-}(\mathbb{R})$ with the families of differential operators can be stated as:
%
\vspace{1.5em}
%
\newline $\bold{Definition}$ $2$: for all $f$ in $\mathbf{s}^{-}(\mathbb{R})$, for all $v\in\mathbb{Z}^+-\{0\}$, for all  $n\in\mathbb{Z}^+$ and $n>1$, the families of operators $(\Psi^{+}_k)_{k \in \mathbb{Z}}$ and $(\Psi^{-}_k)_{k \in \mathbb{Z}}$ ($(\Psi^{+}_k)_{k \in \mathbb{Z}}$ and $(\Psi^{-}_k)_{k \in \mathbb{Z}}$$\subseteq$ $\mathcal{F}(\mathbf{s}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$) decompose uniquely $\partial_t^v$ $f^n$ in $\mathbb{R}$, if for any family of operators $(S_k)_{k \in \mathbb{Z}}$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R})$) decomposing  $\partial_t^v$$f^n$ in $\mathbb{R}$, there exists a unique couple $(\beta_1,\beta_2)$ in $\mathbb{R}^2$ such as: 
%
\begin{equation}
S_k(f) = \beta_1 \Psi^{+}_k(f) + \beta_2 \Psi^{-}_k(f), \qquad \forall k\in\mathbb{Z} 
\end{equation}
%
\vspace{1.5em}
%
%
\\ The main goal of this work is to give a proof of the following lemma and theorem:
%
%
\vspace{1.5em}
%
\\$\bold{Lemma}$: for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, the family of DEO ${\Psi}_{k}^{+}$ ($k=\{0,\pm 1,\pm 2,...\}$) decomposes the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$. 
%
\vspace{1.5em}
%
\\$\bold{Theorem}$: for $f$ in $\mathbf{s}^{-}(\mathbb{R})$, the families of DEO ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ ($k=\{0,\pm 1,\pm 2,...\}$) decompose uniquely the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$. 
%
%\\$\bold{Lemma}$ $2$: In $\mathbf{S}^{-}(\mathbb{R})$, $Im({\Psi}_{k}^{+})$ and $Im({\Psi}_{k}^{-})$ ($k=\{0,\pm 1,\pm 2,...\}$) are a basis of $Im(\partial_t^p f^n)$.
%
\vspace{1.5em}
%
\\ The proofs of the lemma and theorem are given for the $n$-th power of $f$ with $n\in\mathbb{Z}^+$ and $n>1$. A discussion takes place for the special case $n=1$ and $n<0$. In addition, we assume a function $f$ of a variable $t$ with values in $\mathbb{R}$, which can be time or one of the dimension in space ($x,y,z$). 
%To simplify the notation in this manuscript, the variable dependency of the function $f$ is omitted after the second part (i.e. proof of the lemmas).  
%\begin{Theorem}
%
%
%One can define also $Im(\Psi^{-}_k)$ and $Ker(\Psi^{-}_k)$ associated with the family of DEOs $\Psi^{-}_k$ ($k$ in $\mathbb{Z}$). 
%
Note that in Section \ref{SectionProperties}, the study of the properties of the images helps to simplify the formulas shown in Lemma and  Theorem.
%
%The last part is then dedicated to demonstrate the following properties:
%
%\\ $Property$: for $k$ in $\mathbb{Z}$
%\begin{eqnarray}
%Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) &=& \frac{1}{2} Im(\Psi^{+}_k + \Psi^{-}_k) \\
%Ker(\Psi^{+}_k) \bigcap Ker(\Psi^{-}_k) &=& Ker(\Psi^{+}_k + \Psi^{-}_k) \\
%E^{+}_k \bigcap E^{-}_k &=& Ker(\Psi^{+}_k + \Psi^{-}_k)
%\end{eqnarray}
%
\\ Finally,  the last part is dedicated to applying the development in the previous sections to the energy function $\mathcal{E}$ defined as:
%
\begin{equation}\label{EnergyfunDefine02}
\mathcal{E}(f_1(\tau)) = \int_a^{\tau} f_1(t)^2dt < \infty
\end{equation}
%
with $a$ and $\tau$ in $\mathbb{R}$. $f_1$ is assumed to be in $\mathbf{S}^{-}(\mathbb{R})$ and analytic.
% 
%\vspace{1.5em}
%\end{Theorem}
%
%
%
\section{Family of Energy Operators}
%
%
%$f$ is a function continuous in time $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ and $n$ times differentiable($ f\in C^{\infty}$).
%
Following the general description in \cite{Kaiser90}, the general formula of the operator can be written as $P_{-}$ a bilinear form of $\mathbb{R}$ $ \times $ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ defined in the real domain for all functions $f$ and $g$ in $C^{\infty}(\mathbb{R})$ as:
%
%
\begin{equation}
{P}_{-}[f(t),g(t)]=\frac{1}{2}[ f(t)\partial_t g(t)+g(t)\partial_t f(t)]-\frac{1}{2}[f(t)\partial_t^2g(t)+\partial_t^2f(t)g(t)]
\label{Equation Energy Operator in R}
\end{equation}
%
Some years after, the authors in \cite{Maragos1995} introduced the $k$-th differential energy operator (DEO):  
%
\begin{equation}\label{familyDEOdefinition}
{\Psi}_{k}^{-}(f(t)) = \dot{f}(t){f}^{(k-1)}(t)- f(t) {f}^{(k)}(t), \qquad k \in \mathbb{Z}
\end{equation}
%
Note that $k$ is the order of the operators. Here the derivation is following the variable $t$ and ${f}^{(k)}$ means $\partial_t^k f$. One can see that ${\Psi}_{k}^{-}$ is the quadratic form of the bilinear form $P_{-}$.% thus the DEO $\Upsilon_k$ is a bilinear form ($\mathbb{R}$$ \times $ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$).% such as ${\Psi}_{R}^{-}$ in \cite{JPMontillet2010}.
%
%Let us define what means 
To explicitly define ${f}^{(k)}$ for all k in $\mathbb{Z}-\{0\}$:
%
\begin{eqnarray}
f^{(k)}(t) &=& \partial_t^k f(t), \qquad \forall k \in \mathbb{Z}^+ -\{0\} \nonumber \\
f^{(k)}(t) &=& \int_{-\infty}^{t}(\hdots(\int_{-\infty}^{\tau_1}f(\tau_1)d\tau_1)...)d\tau_k, \qquad \forall k \in \mathbb{Z}^- -\{0\}\nonumber \\
f^{(k)}(t) &=& f(t), \qquad  k = 0
\end{eqnarray}
%
With this definition it is important to underline that we are interested in the function such that:
%
\begin{eqnarray}
\partial_t(\int_{-\infty}^{t}f(s)ds) &=& f(t) \nonumber \\
\int_{-\infty}^{t}\partial_{t}f(s)ds &=& f(t)
\end{eqnarray}
%
This explains why we choose $f$ in the Schwartz space $\mathbf{S}^{-}(\mathbb{R})$.
%
%\newline An important property of the DEO shown in \cite{Maragos1995} is the relationship between the orders of the DEO through differentiation (derivative chain rules property) as explained in Equation \eqref{derivative chain rules0111}. 
%
%\begin{equation}\label{rule DEO 1}
%\partial_t {\Psi}_{k}^{-}(f) = {\Psi}_{k+1}^{-}(f) + {\Psi}_{k-1}^{-}(\partial_t f)
%\end{equation}
%
%Let us rename $\Upsilon_k$ has ${\Psi}_{k}^{-}$ for the remaining of this paper.
% In this notation, the $k$-th DEO is the bilinear form \emph{R}$\times$\emph{R} $->$ \emph{R}. 
%
Based on the definition given in \cite{JPMontillet2010}, one can define the DEO family in the same way:
%
\begin{equation}\label{Psik+defdefef}
{\Psi}_{k}^{+}(f) = \dot{f}{f}^{(k-1)} + f {f}^{(k)}, \qquad \forall k \in \mathbb{Z}
\end{equation}
%
${\Psi}_{k}^{+}$ is also a quadratic form with the same properties as ${\Psi}_{k}^{-}$. It is easy to show that the derivative chain rule in Equation \eqref{derivative chain rules0111} remains the same for the DEO family ${\Psi}_{k}^{+}$.
%
%\section{Some Lemmas and Proof of the theorem}\label{LemmasProofs}
%
\section{Proof of the Lemma and Theorem}\label{LemmasProofs}
%
$\bold{Lemma}$: for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, the family of DEO ${\Psi}_{k}^{+}$ ($k=\{0,\pm 1,\pm 2,...\}$) decomposes the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$. 
%
\begin{proof}
%
The general proof is structured via an induction for the existence of the decomposition. First, the study of the four first derivatives of some selected powers of $f$ ($n=\{2, 3\}$) shows how the differential energy operators (or DEOs) are defined for each selected power of $f$. Furthermore, it gives a the method to find the energy operator family to decompose the successive derivatives. The case $n=p$ ($p>1$) ends the induction proof.
%is established through the first $4$ derivatives and induction. 
%Note that for the case $n=\{2,3\}$ a general formula is given to decompose $\partial_t^v f^n$ ($ v \in \mathbb{Z}^+-\{0\}$).
The non-uniqueness of the decomposition is justified in a separate paragraph with a counter example.
%
\vspace{1.5em}
%
\newline A- $\bold{Existence}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$ $\bold{of}$ $\bold{the}$ $\bold{Power}$ $\bold{of}$ $f$ $\bold{via}$ $\bold{DEOs}$
%
\vspace{1.5em}
%
\\ $\bold{Case}$ $n=2$:
%
\vspace{1.5em}
%
\\Let $f$ be a function in $\mathbf{S}^{-}(\mathbb{R})$. %It is then possible to express $(f)^2$ in Taylor - Mc Laurain series such as:
%
%\begin{equation}
%{f(t)}^2 = \Sigma_{k=0}^{\infty} (\partial_t^k f(0)^2) \frac{t^k}{k!}, \qquad \forall t \in \emph{R} 
%\end{equation}
%
One can write the coefficient $\partial_t^v f^2$ for $v=\{1,2,3,4\}$ as:
%
\begin{eqnarray}\label{Coeff2n}
%
\partial_t f^2 &=& 2 f \partial_t f \nonumber \\
\partial_t f^2 &=& {\Psi}_{1}^{+}(f) \nonumber \\
& & \nonumber \\
\partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\
\partial_t^2 f^2 &=& \partial_t {\Psi}_{1}^{+}(f) \nonumber \\
\partial_t^2 f^2 &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) \nonumber \\
& & \nonumber \\
\partial_t^3 f^2 &=& {\Psi}_{3}^{+}(f) +2 {\Psi}_{1}^{+}(\partial_t f) + {\Psi}_{-1}^{+}(\partial_t^2 f) \nonumber \\
& & \nonumber \\
\partial_t^4 f^2 &=& {\Psi}_{4}^{+}(f) +3 {\Psi}_{2}^{+}(\partial_t f) +3 {\Psi}_{0}^{+}(\partial_t^2 f) + {\Psi}_{-2}^{+}(\partial_t^3 f) \nonumber \\
%
\end{eqnarray}
%
%For the case $n=2$, the above development of the coefficients of the Taylor series can be factorized directly with the energy operators. 
In this example, the successive coefficients are calculated using the derivative chain rule (e.g. Equation \eqref{derivative chain rules0111}). The scalars for each coefficient follow the Pascal's triangle rule and the order of the derivatives. Moreover, it is possible to develop the same type of Pascal's triangle rule to predict not only the scalar coefficients but also the order of the energy operators ($k$) involved for each derivative. 
%
%
\begin{figure}[!hbp]
\begin{center}
%\includegraphics[width=2in\textwidth]{extra/unw_tf_275}
\includegraphics[width=5in]{essai_pascales01}
\caption{\footnotesize{Pascal's triangle rule for the energy operators}}\label{Pascaletrainglerules}
\end{center}
\end{figure}
%
%
Figure \ref{Pascaletrainglerules} summarizes this general rule. It is straightforward to see that for the $v$-th derivative order, the highest DEO order is equal to $v$. Then, the remaining energy operator orders are calculated  by decreasing the order by one for each previous  energy operator involved in the ($v-1$)-th derivative. Secondly, it is important to see that all the DEOs involved to approximate the $v$-th derivative of $f^2$ are applied to the $s$-th derivatives of $f$ with $s=\{1,...,v-1\}$, except the DEO with the highest order (which is applied directly to $f$).  %forgetting to increase the order of the derivative of $f$ (i.e. from $p$ to $p+1$) in ${\Psi}_{k}^{+}(\partial_t^p f)$.
%
For simplicity using Equation \eqref{Coeff2n}, one can write ($\forall s \in \{1,2,...,m\}$):
%
\begin{eqnarray}\label{Aocorf1N2}
 a_s^+(f) &=& \partial_t^s f^2 \nonumber \\
a_s^+(f) &=& \partial_t^{s-1} \Psi_{1}^{+} (f) 
\end{eqnarray}
%
$a_s^+$ is a sum of DEOs. The generalization of $a_s^+(f)$ is given via the formula:
%
%\begin{eqnarray}\label{aidefderivate1}
%
%\partial_t a_k^+ (f) &=& a_{k+1}^+(f), \qquad k=\{0,\pm 1,\pm 2,...\}
%
%\end{eqnarray}
%
%
%A further study of the derivatives of $f^2$ ends up with the general formula based on the Pascal's triangle rules (see \cite{Kreyszig}) such as:
%
\begin{eqnarray}\label{coefficientsAPgeneral}
a_s^+(f) &=& \sum_{k=0}^{s-1} \big(_{k}^{s-1} \big) \Psi_{2(k+1)-s}^{+}(\partial_t^{s-k-1}f), \qquad \forall s\in \mathbb{Z}^{+}-\{0\} 
\end{eqnarray}
%
This formula can be checked for $s=\{1,2,3,4\}$ using Equation \eqref{Coeff2n}.
 %It is difficult to demonstrate it by induction.
Let us assume the formula is true for $s=m$. One can write:
%
\begin{eqnarray}\label{coefficientsAPgeneralbis}
a_{m}^+(f) &=& \partial_t a_{m-1}^+(f)\nonumber \\
a_{m}^+(f) &=& \partial_t \sum_{k=0}^{m-2} \big(_{k}^{m-2} \big) \Psi_{2(k+1)-m+1}^{+}(\partial_t^{m-k-2}f)\nonumber \\
a_{m}^+(f) &=& \sum_{k=0}^{m-2} \big(_{k}^{m-2} \big) [ \Psi_{2(k+1)-m+2}^{+}(\partial_t^{m-k-2}f) + \Psi_{2(k+1)-m}^{+}(\partial_t^{m-k-1}f)]
\nonumber \\
%a_{p+1}^+(f) &=& \sum_{k=0}^{p-1} \big(_{k}^{p-1} \big) \partial_t \Psi_{2(k+1)-p}^{+}(\partial_t^{p-k-1}f) \nonumber \\
%a_{p+1}^+(f) &=& \sum_{k=0}^{p-1} \big(_{k}^{p-1} \big) [ \Psi_{2(k+1)-p+1}^{+}(\partial_t^{p-k-1}f) + \Psi_{2(k+1)-p-1}^{+}(\partial_t^{p-k}f)] \nonumber \\
\end{eqnarray}
%
Now using Equations \eqref{coefficientsAPgeneralbis} and the derivative chain rule property, the case $m+1$ is:
%
%%%-> here I am :)
%
\begin{eqnarray}\label{coefficientsAPgeneralbis2}
a_{m+1}^+(f) &=& \partial_t a_{m}^+(f)\nonumber \\
a_{m+1}^+(f) &=& \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [ \Psi_{2(k+1)-m+1}^{+}(\partial_t^{m-k-1}f) + \Psi_{2(k+1)-m-1}^{+}(\partial_t^{m-k}f)] \nonumber \\
%a_{p+1}^+(f) &=& \sum_{k=0}^{P} \big(_{k}^{P} \big) \Psi_{2(k+1)-P-1}^{+}(\partial_t^{P-k}f) \nonumber \nonumber \\
a_{m+1}^+(f) &=& \sum_{k=0}^{m} \big(_{k}^{m} \big) \Psi_{2(k+1)-m-1}^{+}(\partial_t^{m-k}f)\nonumber \\
\end{eqnarray}
%
The above equations finish the induction proof for this particular case.% the formula in Equation \eqref{coefficientsAPgeneral} is demonstrated.
% 
% Let us assume it works for $p=n$. Now for the case $p=n+1$, one can write:
%
%
%$\mathbb{Z}^{+}$ is the set of the positive integer values including $0$. Thus, this process of decomposing the coefficients of the Taylor- Mc Laurin series with the DEOs shows that the family of energy operator ${\Psi}_{k}^{+}$ forms a base generator for $f^2$. Note that ${\Psi}_{1}^{-}$, the first order of the DEO ${\Psi}_{k}^{-}$, is equal to the energy operator null (\emph{R}$\times$\emph{R} $\rightarrow$ $0$).
%
%
\vspace{1.5em}
%
 \newline $\bold{Case}$ $n=3$:
%
\vspace{1.5em}
%
\\ In the same way as in the previous case, we write the development of the first coefficient:% of the Taylor -Mc Laurin Series.
%
\begin{eqnarray}\label{f3coef1}
%
\partial_t f^3 &=& 3 f^2 \partial_t f \nonumber \\
%
\end{eqnarray}
%
Let us introduce a DEO which is just a product of ${\Psi}_{k}^{+}$ by a constant as:
%
\begin{equation}\label{GammaOperatorfamily}
%\Gamma_{k}^{+} (f) = 2 \dot{f_t}{f_t}^{k-1} + f_t {f_t}^{k} \qquad k=\{0,\pm 1,\pm 2,...\}
\Gamma_{k}^{+} (f) = \frac{3}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}), \qquad \forall k\in \mathbb{Z}
\end{equation}
%
Clearly, by definition $\Gamma_{k}^{+}$ is a quadratic form and a DEO as it is proportional to ${\Psi}_{k}^{+}$. Note that the derivative properties shown in Equation \eqref{derivative chain rules0111} hold for this DEO. Using Equation \eqref{Aocorf1N2}, it is possible to write a similar equality with $a_k^{+}:$  $A_i^+(f) = 3/2 \partial_t^{i-1} a_i^+(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). It is then possible to write the Equation \eqref{f3coef1} and the successive derivatives of $f^3$ as :
%
\begin{eqnarray}
%
\partial_t f^3 &=& f \Gamma_{1}^{+}(f) \\
\partial_t f^3 &=& f  A_1^{+}(f) \nonumber \\
%
 & & \nonumber \\
%
\partial^2_t f^3 &=& f (\Gamma_{2}^{+}(f) + \Gamma_{0}^{+}(\partial_t f)) + \partial_t f \Gamma_{1}^{+} (f) \nonumber \\
%
\partial^2_t f^3 &=& f A_2^{+}(f)  + \partial_t f A_1^{+}(f) \\
%
 & & \nonumber \\
%
\partial^3_t f^3 &=& f (\Gamma_{3}^{+}(f) + 2 \Gamma_{1}^{+}(\partial_t f) + \Gamma_{-1}^{+}(\partial^2_t f)) + 2 \partial_t f (\Gamma_{2}^{+}(f) + \Gamma_{0}^{+}(\partial_t f)) + \partial^2_t f \Gamma_{1}^{+} (f) \nonumber \\
%
\partial^3_t f^3 &=& f A_3^{+}(f) + 2 \partial_t f A_2^{+}(f) + \partial^2_t f A_1^{+}(f) \\
%
& & \nonumber \\
%
\partial^4_t f^3 &=& f (\Gamma_{4}^{+}(f) + 3 \Gamma_{2}^{+}(\partial_t f) + 3 \Gamma_{0}^{+}(\partial^2_t f) + \Gamma_{-2}^{+}(\partial^2_t f)) \nonumber \\
& & + 3 \partial_t f (\Gamma_{3}^{+}(f) + 2 \Gamma_{1}^{+}(\partial_t f) + \nonumber \\
 && \Gamma_{-1}^{+}(\partial^2_t f)) + 3 \partial^2_t f (\Gamma_{2}^{+}(f) + \nonumber \\ 
 & & \Gamma_{0}^{+}(\partial_t f)) + \partial^3_t f \Gamma_{1}^{+} (f) \nonumber \\
%
\partial^4_t f^3 &=& f A_4^{+}(f) + 3 \partial_t f A_3^{+}(f) + 3 \partial^2_t f A_2^{+}(f) + \partial^3_t f A_1^{+}(f) \nonumber \\
%
\end{eqnarray}
%
%Note that $a_k^+(f)$ is equal to $3/2$$a_k^+(f)$ as defined in Equation \eqref{coefficientsAPgeneral}.
%
\\ There is a certain symmetry between the above equations and Equations \eqref{Coeff2n}. The decomposition of $\partial_t^s f^3$ is performed using the DEO $\Gamma_{k}^{+}$. Using the equations above and the general definition of $A_s^{+}$ ($\forall s \in \mathbb{Z}^+ -\{0\}$) in Equation \eqref{coefficientsAPgeneral}, it is then possible to generalize the formula: 
%
\begin{equation}\label{dtp3fwithAp}
\partial_t^{m+1} f^3 =\sum_{k=0}^m \big(_{k}^m \big) A_{k+1}^+(f) \partial_t^{m-k} f, \qquad \forall m\in \mathbb{Z}^+
\end{equation}
%
This formula is checked with $m=\{0,1,2,3\}$. In order to finish the proof by induction, let us then assume the formula true for $m+1$. Following Equation \eqref{dtp3fwithAp} and the previous development, one can write:
%
\begin{eqnarray}\label{dtp3fwithAp1b}
\partial_t^{m+1} f^3 &=& \partial_t (\partial_t^{m} f^3)\nonumber \\
\partial_t^{m+1} f^3 &=&\sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [A_{k+2}^+(f) \partial_t^{m-k-1} f +A_{k+1}^+(f) \partial_t^{m-k} f]
\end{eqnarray}
%
%
Finally using Equation \eqref{dtp3fwithAp1b}, the development for  $m+2$ is:
%
\begin{eqnarray}\label{dtp3fwithAp02}
\partial_t^{m+2} f^3 &=& \partial_t(\partial_t^{m+1} f^3)\nonumber \\
\partial_t^{m+2} f^3 &=& \partial_t (\sum_{k=0}^m \big(_{k}^m \big) A_{k+1}^+(f) \partial_t^{m-k} f) \nonumber \\
\partial_t^{m+2} f^3 &=&  \sum_{k=0}^m \big(_{k}^m \big) [A_{k+2}^+(f) \partial_t^{m-k} f +A_{k+1}^+(f) \partial_t^{m-k+1} f] \nonumber \\
\partial_t^{m+2} f^3 &=& \sum_{k=0}^{m+1} \big(_{k}^{m+1} \big) A_{k+1}^+(f) \partial_t^{m-k+1} f \nonumber \\
\end{eqnarray}
%
This ends the induction proof to show Equation \eqref{dtp3fwithAp}, and also this particular case.
%Now if we change $p=P+1$, the above formula is equal to:
%
%\begin{eqnarray}\label{dtp3fwithAp03}
% \partial_t^{p+2} f^3 &=&  \sum_{k=0}^{P+1} \big(_{k}^{P+1} \big) [A_{k+2}^+(f) \partial_t^{P-k+1} f +A_{k+1}^+(f) 
%
%\begin{equation}\label{equation042b}
%\forall k\in \mathbb{Z}^+-\{0\}, \qquad \partial^k_t f^4 = \partial^{k-1}_t (f^2 a_1^+(f))
%\end{equation}
%%
%
%%
\vspace{1.5em}
%
\\ $\bold{Case}$ $n=p$ with $p>1$:
%
\vspace{1.5em}
%
\\ First, let us assume that the family of DEO ${\theta}_{k}^{+}$ ($k\in\mathbb{Z}$) proportional to the family of DEO  ${\Psi}_{k}^{+}$ ($k\in\mathbb{Z}$) which decomposes the successive derivatives of ${f}^{p-1}$ with the definition:
%
\begin{eqnarray}
{\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)})
\end{eqnarray}
%
% As done in the previous case, let us develop the first derivative for ${f}^{p}$:
In addition, one can define  $B_i^+(f) = \partial_t^{i-1} {\theta}_1^+(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). Let us then write the first derivatives of ${f}^{p}$:
%
\begin{eqnarray}\label{NONOVO}
 \partial_t f^p &=&  p f^{p-1} \partial_t f  \nonumber \\
 \partial_t f^p &=&  \frac{p}{2} f^{p-2} \Psi_1^+(f)  \nonumber \\
\partial_t f^p &=& \frac{p}{p-1} {\theta}_{1}^{+}(f) f^{p-2} \nonumber \\
\partial_t f^p &=& \frac{p}{p-1} B_1^+(f)  f^{p-2}\nonumber \\
%
& & \nonumber \\
%
%\partial_t^2 f^p &=& \frac{p(p-2)}{p-1} {\theta}_{1}^{+}(f) f^{p-3} \partial_t f + \frac{p}{p-1} ({\theta}_{2}^{+}(f) + {\theta}_{0}^{+}( \partial_t f)) f^{p-2}   \nonumber \\
\partial_t^2 f^p &=& \frac{p}{p-1} B_1^+(f) \partial_t f^{p-2}+ \frac{p}{p-1} B_2^+(f)  f^{p-2} \nonumber \\
%\partial_t^3 f^p &=&   \frac{p(p-2)(p-3)}{p-1} {\theta}_{1}^{+}(f) f^{p-4} \partial_t f+ 2\frac{p(p-2)}{p-1} ({\theta}_{2}^{+}(f) + {\theta}_{0}^{+}( \partial_t f)) f^{p-3} \nonumber \\
%& & + \frac{p}{p-1} ({\theta}_{3}^{+}(f) + 2{\theta}_{1}^{+}( \partial_t f) +{\theta}_{-1}^{+}( \partial_t^2 f)) f^{p-2}\nonumber \\
\partial_t^3 f^p &=& \frac{p}{p-1}  B_1^+(f) \partial_t^2 f^{p-2}+ 2\frac{p}{p-1}B_2^+(f) \partial_t f^{p-2} \nonumber \\
& & + \frac{p}{p-1}B_3^+(f) f^{p-2}
\end{eqnarray}
%
As shown in the previous case, we can generalize the formula for the $s$-th derivative as:
%
\begin{equation}\label{dtppfwithAps}
\partial_t^{s+1} f^p =\sum_{k=0}^s \big(_{k}^s \big) \frac{p}{p-1}B_{k+1}^+(f) \partial_t^{s-k} f^{p-2}, \qquad \forall s \in \mathbb{Z}^+
\end{equation}
%
The formula in Equation \eqref{dtppfwithAps} has just been verified for $s=\{0,1,2\}$. Furthermore, one can write for the case $s=m$:
%
\begin{eqnarray}\label{dtp3fwithAp1b}
\partial_t^{m+1} f^p &=& \partial_t (\partial_t^{m} f^p) \nonumber  \\
\partial_t^{m+1} f^p &=&\frac{p}{p-1} \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [B_{k+2}^+(f) \partial_t^{m-k-1} f^{p-2} + B_{k+1}^+(f) \partial_t^{m-k} f^{p-2}] \nonumber  \\
\end{eqnarray}
%
%
Let us assume that the Equation \eqref{dtppfwithAps} is true for $s=m+1$. Using the previous equation, the case $s=m+2$ is developed as:
%
\begin{eqnarray}\label{dtp3fwithAp02}
\partial_t^{m+2} f^p &=& \frac{p}{p-1} \partial_t (\sum_{k=0}^m \big(_{k}^m \big) B_{k+1}^+(f) \partial_t^{m-k} f^{p-2}) \nonumber \\
\partial_t^{m+2} f^p &=&  \frac{p}{p-1} \sum_{k=0}^v \big(_{k}^m \big) [B_{k+2}^+(f) \partial_t^{m-k}  f^{p-2} +B_{k+1}^+(f) \partial_t^{m-k+1} f^{p-2}] \nonumber \\
\partial_t^{m+2} f^p &=& \frac{p}{p-1} \sum_{k=0}^{m+1} \big(_{k}^{m+1} \big) B_{k+1}^+(f) \partial_t^{m-k+1} f^{p-2} \nonumber \\
\end{eqnarray}
%
$(B_k^+)_{k\in  \mathbb{Z}}$ and by definition, $(\theta_k^+)_{k\in  \mathbb{Z}}$ decompose $\partial_t^{s} f^p$  ($s \in \mathbb{Z}^+$, $p\in \mathbb{Z}^+ -\{0,1\}$). This ends the induction proof to confirm Equation \eqref{dtppfwithAps}. 
%\begin{eqnarray}\label{equationp01}
%\partial_t f^p &=& \frac{{\psi}_{1}^{+}(f)}{p-1}f^{p-2}+ f f^{p-3}{\psi}_{1}^{+}(f) \nonumber \\
%\partial_t f^p &=& \big ( 1+\frac{1}{p-1} \big )f^{p-2}{\psi}_{1}^{+}(f) \nonumber \\
%\partial_t f^p &=& f^{p-2}{\beta}_{1}^{+}(f), \qquad \forall p \in \mathbb{Z}^+, \qquad p > 1
%\end{eqnarray}
% 
%quation (\ref{equationp01}) shows the existence of a DEO family ${\beta}_{k}^{+}$ ($k\in\mathbb{Z}$) proportional to ${\psi}_{k}^{+}$ (which is proportional to ${\Psi}_{k}^{+}$ by definition).
%
%
%
%
Finally as ${\theta}_{k}^{+}$ is proportional to ${\Psi}_{k}^{+}$, one can conclude that the family of DEO  ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$) decomposes the successive derivatives of ${f}^{p}$ ($p \in \mathbb{Z}^+$, and $p> 1$). This ends the induction proof on the $n$-th power of $f$. 
%
%
%%
%\newline Finally, the recurrence on the derivatives of $f^p$ ($p>1$) shows that it is possible to use the DEO family introduced in Equation \eqref{rule DEO 1} to decompose the $n$-th derivative of $f^p$ ($\partial^n_t f^p$) with $p>1$.
%%  
\vspace{1.5em}
%
\newline B -$\bold{Uniqueness}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$
%
\\ In the previous proof, it has been shown that given a family of DEOs  ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$), it is possible to decompose the successive derivatives $\partial_t^i f^n$ ($n\in \mathbb{Z}^+$, $n>1$, $i\in\mathbb{Z}^+$).
%
However, there is no uniqueness of the decomposition of $f^n$ with the DEO family ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$). A simple counter example can be found using the DEO family:
%
\begin{equation}\label{definitionOfeta}
 \eta_k(f) = 3(\partial_t f \partial_t^{k-1} f)-f\partial_t^k f , \qquad \forall k \in \mathbb{Z}
\end{equation}
%
Note that the derivative chain rule property is applied to this operator. One can verify:
%
 \begin{eqnarray}\label{Coeff2nnn}
%
\partial_t f^2 &=& 2 f \partial_t f \nonumber \\
\partial_t f^2 &=& \eta_1(f) \nonumber \\ 
\eta_1(f)&=&{\Psi}_{1}^{+}(f) \nonumber \\
& & \nonumber \\
\partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\
\partial_t^2 f^2 &=&  \partial_t {\eta}_{1}^{+}(f) \nonumber \\
\partial_t {\Psi}_{1}^{+}(f) &=& \partial_t {\eta}_{1}^{+}(f) \nonumber \\
\partial_t^2 f^2 &=& {\eta}_{2}^{+}(f) + {\eta}_{0}^{+}(\partial_t f) \nonumber \\
& & \nonumber \\
\partial_t^3 f^2 &=&  \partial_t^2 {\eta}_{1}^{+}(f) \nonumber \\
\partial_t^2 {\Psi}_{1}^{+}(f) &=& \partial_t^2 {\eta}_{1}^{+}(f) \nonumber \\
\partial_t^3 f^2 &=& {\eta}_{3}^{+}(f) +2 {\eta}_{1}^{+}(\partial_t f) + {\eta}_{-1}^{+}(\partial_t^2 f) \nonumber \\
& & \nonumber \\
\partial_t^4 f^2 &=&  \partial_t^3 {\eta}_{1}^{+}(f) \nonumber \\
\partial_t^3 {\Psi}_{1}^{+}(f) &=& \partial_t^3 {\eta}_{1}^{+}(f) \nonumber \\
\partial_t^4 f^2 &=& {\eta}_{4}^{+}(f) +3 {\eta}_{2}^{+}(\partial_t f) +3 {\eta}_{0}^{+}(\partial_t^2 f) + {\eta}_{-2}^{+}(\partial_t^3 f) 
%
\end{eqnarray}
% 
%
%
%
\end{proof}
%
%
It is important to underline the fact that $(a_k^+)_{k\in\mathbb{Z}^+-\{0\}}$, $(A_k^+)_{k\in\mathbb{Z}^+-\{0\}}$ and $(B_k^+)_{k\in\mathbb{Z}^+-\{0\}}$ are not included in the definitions $1$ and $2$ as they do not follow the derivative chain rule.  In addition, $Definition$ $1$ can also be simplified fixing $l=2$ based on the definition of the energy operator in Equation \eqref{Psik+defdefef}.
%
\vspace{1.5em}
%
\newline $\bold{Theorem}$: for $f$ in $\mathbf{s}^{-}(\mathbb{R})$, the family of DEOs ${\Psi}_{k}^{-}$  and ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$) decomposes uniquely the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$. 
%
%
\begin{proof}
%
The proof is separated into three parts. The preliminary part justifies by induction the decomposition of the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$ with ${\Psi}_{k}^{-}$  and ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$). The proof is similar to the one in the Lemma, hence some parts are shorten to avoid long repetitions.  
\\The second section focuses on the existence of the decomposition as defined in Definition $1$ and Definition $2$. This is also shown by induction on the successive derivatives of $f^n$ ($n\in\mathbb{Z}^+$ and $n>1$) reusing the examples of different families of energy operator previously seen. Finally, the uniqueness is detailed in the last part.
%
\vspace{1.5em}
%
\newline A- $\bold{Preliminary}$
%
\vspace{1.5em}
%
\newline Recall the formula of ${\Psi}_{1}^{-}(f)$:
%
\begin{equation}
{\Psi}_{1}^{-}(f) = \dot{f}f - f \dot{f} =0
\end{equation}
%
One can see from the definition of ${\Psi}_{1}^{-}$, that this DEO projects any smooth real-valued function $f$ (in $\mathbf{s}^{-}(\mathbb{R})$) onto the null space (${\Psi}_{1}^{-}(f)$: $\mathbb{R}\times\mathbb{R}$ $\rightarrow$ $0$).
%
For any $f$ in $\mathbf{s}^{-}(\mathbb{R})$,
%
\vspace{1.5em}
%
\newline $\bold{Case}$ $n=2$:
%
\vspace{1.5em}
% 
%
%
\begin{eqnarray}\label{equationPsikm01}
\partial_t f^2 &=& f \partial_t f + f\partial_t f + f\partial_t f - f\partial_t f \nonumber \\
\partial_t f^2 &=& {\Psi}_{1}^{+}(f) + {\Psi}_{1}^{-}(f)\nonumber \\
& & \nonumber \\
\partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\
\partial_t^2 f^2 &=& \partial_t ({\Psi}_{1}^{+}(f) +{\Psi}_{1}^{-}(f)) \nonumber \\
\partial_t^2 f^2 &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) + \nonumber \\
                 &&   {\Psi}_{2}^{-}(f) + {\Psi}_{0}^{-}(\partial_t f) \nonumber \\
& & \nonumber \\
\partial_t^3 f^2 &=& ({\Psi}_{3}^{+}(f) + {\Psi}_{3}^{-}(f)) +2 ({\Psi}_{1}^{+}(\partial_t f)+{\Psi}_{1}^{-}(\partial_t f)) + \nonumber \\
& & ({\Psi}_{-1}^{+}(\partial_t^2 f) +{\Psi}_{-1}^{-}(\partial_t^2 f)) \nonumber \\
\end{eqnarray}
%
In this development, there is a symmetry with the proof of the previous lemma (e.g., Equation \eqref{Coeff2n}). Thus, this allows to shorten the proof by induction of the decomposition of the successive derivatives of the $n$-th power of $f$. Therefore, a necessary step as shown in the demonstration of the existence in the Lemma, is the demonstration of a similar formula for $\partial_t^k ({\Psi}_{1}^{-}(f))$ (e.g., Equation \eqref{coefficientsAPgeneralbis2}). It ends with the generalization for $n=p$.% The uniqueness is shown in a separated paragraph. 
%
%\newline First, one can see from  the definition of ${\Psi}_{1}^{-}$ in Equation \eqref{familyDEOdefinition}, this DEO projects any smooth real-valued function $f$ (in $\mathbf{S}^{-}(\mathbb{R})$) onto the null space (${\Psi}_{1}^{-}(f)$: $\mathbb{R}\times\mathbb{R}$ $\rightarrow$ $0$).% for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$. %Based on this result, the proof is straightforward.
%
\\ From Equations \eqref{equationPsikm01}, one can define $a_s^-(f)$ in the same way that $a_s^+(f)$ was defined in Equations \eqref{Aocorf1N2} as:
%
\begin{equation}\label{Aocorf1N3}
a_s^-(f) = \partial_t^{s-1} \Psi_{1}^{-} (f), \qquad \forall s\in \mathbb{Z}^+-\{0\}
\end{equation}
%
%In the following, let us assume that $f$ is in $Ker(\Psi^{+}_j)$ ($j$ in $\mathbb{Z}$). One can write:
%
%\begin{equation}\label{Coeff2nmoinsmm}
%
With the property of the derivative chain rule in Equation \eqref{derivative chain rules0111} and Equation \eqref{Aocorf1N2}, it is easy to calculate the first terms of the DEO family $a_s^-(f)$ such as :
%
%
\begin{eqnarray}\label{Coeff2nmoins}
%
%\partial_t f^2 &=& 2 f \partial_t f \nonumber \\
a_1^-(f)={\Psi}_{1}^{-}(f) &=& 0 \nonumber \\
& & \nonumber \\
a_2^-(f)=\partial_t {\Psi}_{1}^{-}(f) &=& {\Psi}_{2}^{-}(f) + {\Psi}_{0}^{-}(\partial_t f) \nonumber \\
{\Psi}_{2}^{-}(f) &=& -{\Psi}_{0}^{-}(\partial_t f) \nonumber \\
& & \nonumber \\
a_3^-(f)=\partial_t^2 {\Psi}_{1}^{-}(f) &=& {\Psi}_{3}^{-}(f) +2 {\Psi}_{1}^{-}(\partial_t f) + {\Psi}_{-1}^{-}(\partial_t^2 f) \nonumber \\
{\Psi}_{3}^{-}(f) &=& -{\Psi}_{-1}^{-}(\partial_t^2 f) \nonumber \\
& & \nonumber \\
a_4^-(f)=\partial_t^3 {\Psi}_{1}^{-}(f) &=& {\Psi}_{4}^{-}(f) +3 {\Psi}_{2}^{-}(\partial_t f) +3 {\Psi}_{0}^{-}(\partial_t^2 f) + {\Psi}_{-2}^{-}(\partial_t^3 f) \nonumber \\
{\Psi}_{4}^{-}(f) &=& -{\Psi}_{-2}^{-}(\partial_t^3 f)
%
\end{eqnarray}
%
The above equations are similar to the development in Equation \eqref{Coeff2n}. % Thus, one can deduce a similar formula for $a_k^-(f)$.
%
%
In addition, the family of DEO ${\Psi}_{k}^{-}$ ($k\in\mathbb{Z}$) has  the same derivative properties as ${\Psi}_{k}^{+}$. A similar equation can then be established for $a_s^-(f)$ following the development written in Equation \eqref{Coeff2n} as:
%
\begin{eqnarray}\label{coefficientsAPgeneral2bis}
a_s^-(f) &=& \sum_{k=0}^{s-1} \big(_{k}^{s-1} \big) \Psi_{2(k+1)-s}^{-}(\partial_t^{s-k-1}f), \forall s\in \mathbb{Z}^{+} -\{0\}
\end{eqnarray}
%
and, 
%
\begin{equation}
 a_s^-(f) = \partial_t^s \Psi_1^{-}(f) =0, \qquad \forall s \in \mathbb{Z}^+-\{0\}, \qquad \forall f\in\mathbf{s}^{-}(\mathbb{R})  
\end{equation}
%
This formula has just been checked for $s=\{1,2,3,4\}$ with the Equation \eqref{Coeff2nmoins}.
%
The generalization of the formula for $s=m$ is very similar to that described in the Equations \eqref{coefficientsAPgeneralbis} literally by changing $+$ and $-$ in the definition of the energy operator. It follows that the decomposition of the successive derivatives of $f^2$ is generalized for any $p$ in $\mathbb{Z}^+-\{0\}$ as:
%
\begin{eqnarray}\label{Coeff2nmoinspp}
\partial_t^m f^2 &=& \partial_t^{m-1} ({\Psi}_{1}^{+}(f) +{\Psi}_{1}^{-}(f)) \nonumber \\
                 %&=& a_{p-1}^+(f) + a_{p-1}^-(f) \nonumber \\
                 &=& \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) \Psi_{2(k+1)-m}^{+}(\partial_t^{m-k-1}f) + \nonumber \\
                 & & \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) \Psi_{2(k+1)-m}^{-}(\partial_t^{m-k-1}f) \nonumber \\
\end{eqnarray}
%
In this case, $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$  and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ directly decompose $\partial_t^m f^2$. Note that the family $(a_k^-)_{k\in\mathbb{Z}^+-\{0\}}$ does not follow the derivative chain rule, and thus cannot be defined using definitions $1$ and $2$. In addition, the Equation \eqref{coefficientsAPgeneral2bis} can be easily extended for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$ if we follow the induction proof in the Lemma (e.g., Equations \eqref{coefficientsAPgeneralbis}) as there is no restriction on the Images of the DEOs $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$.
%
\vspace{1.5em}
%
\\$\bold{Case}$ $n=p$ with $p>1$: 
%
\vspace{1.5em}
%
\\ Following the same step as in the proof of the previous lemma, let us define the families of DEO ${\theta}_{k}^{-}$ proportional to the family of DEO  ${\Psi}_{k}^{-}$  and ${\theta}_{k}^{+}$ proportional to the family of DEO  ${\Psi}_{k}^{+}$ ($k\in\mathbb{Z}$) with the assumption that they decompose the successive derivatives of ${f}^{p-1}$ as:
%
\begin{eqnarray}\label{psi+0008}
{\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\
&= & \frac{(p-1)}{2} \Psi_k^+(f) \nonumber \\
{\theta}_{k}^{-}(f) &= & \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} - f {f}^{(k)}) \nonumber \\
&= &\frac{(p-1)}{2} \Psi_k^-(f)
\end{eqnarray}
%
%Let us then write the first derivative of ${f}^{p}$ such as:
Following the same development as in Equation \eqref{NONOVO}, one can define  $B_i^+(f) = \partial_t^{i-1} {\theta}_1^+(f)$ and $B_i^-(f) = \partial_t^{i-1} {\theta}_1^-(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). Let us then write the first derivatives of ${f}^{p}$ such as:
%
\begin{eqnarray}\label{NONOVO2}
 \partial_t f^p &=& p f^{p-1} \partial_t f  \nonumber \\
  \partial_t f^p &=& \frac{p}{2} f^{p-2} (\Psi_k^+(f) +\Psi_k^-(f))  \nonumber \\
 \partial_t f^p &=& \frac{p}{p-1} ({\theta}_{1}^{+}(f)+{\theta}_{1}^{-}(f)) f^{p-2} \nonumber \\
\partial_t f^p &=& \frac{p}{p-1} (B_1^+(f) + B_1^-(f))  f^{p-2} \nonumber \\
%
& & \nonumber \\
%
%\partial_t^2 f^p &=& \frac{p(p-2)}{p-1} ({\theta}_{1}^{+}(f)+{\theta}_{1}^{-}(f)) f^{p-3} + \frac{p}{p-1} ({\theta}_{2}^{+}(f) + {\theta}_{0}^{+}( \partial_t f) +{\theta}_{2}^{-}(f) + {\theta}_{0}^{-}( \partial_t f)) f^{p-2}  \nonumber \\
\partial_t^2 f^p &=&  \frac{p}{p-1} (B_1^+(f)+B_1^-(f)) \partial_t f^{p-2}+ \frac{p}{p-1} (B_2^+(f)+B_2^-(f))  f^{p-2} )\nonumber \\
%\partial_t^3 f^p &=&   \frac{p(p-2)(p-3)}{p-1} {\theta}_{1}^{+}(f) f^{p-4} + 2\frac{p(p-2)}{p-1} ({\theta}_{2}^{+}(f) + {\theta}_{0}^{+}( \partial_t f)) f^{p-3} \nonumber \\
%& & + \frac{p}{p-1} ({\theta}_{3}^{+}(f) + 2{\theta}_{1}^{+}( \partial_t f) +{\theta}_{-1}^{+}( \partial_t^2 f)) f^{p-2}\nonumber \\
%\partial_t^3 f^p &=& \frac{p(p-2)(p-3)}{p-1}  B_1^+(f) f^{p-4}+ 2\frac{p(p-2)}{p-1}B_2^+(f) f^{p-3} \nonumber \\
%& & + \frac{p}{p-1}B_3^+(f) f^{p-2}
\end{eqnarray}
%
There is again a symmetry with the proof in the lemma. Following Equation \eqref{dtppfwithAps}, we can define the $s+1$-th derivative of $f^p$ using $B_{k+1}^-(f)$ and $B_{k+1}^+(f)$: 
%
\begin{equation}\label{dtppfwithApsd2}
\partial_t^{s+1} f^p = \sum_{k=0}^s \big(_{k}^s \big) \frac{p}{p-1}(B_{k+1}^-(f) +B_{k+1}^+(f)) \partial_t^{s-k} f^{p-2}, \qquad \forall s \in \mathbb{Z}^+
\end{equation}
%
%
This equation has just been checked for $s=\{0,1\}$. As the induction proof follows exactly the same development as in Equation \eqref{dtp3fwithAp02} by only adding $B_{k+1}^-(f)$ (with the same properties as $B_{k+1}^+(f)$ in Equation \eqref{NONOVO2}), it allows then to assume the generalization to the case $s+2$.
%
\newline Thus,  $(B_{k}^+)_{k\in\mathbb{Z}}$ and  $(B_{k}^-)_{k\in\mathbb{Z}}$ decompose the $s$-th derivative of $f^p$. From their definition, one can conclude that $(\theta_{k}^+)_{k\in\mathbb{Z}}$ and $(\theta_{k}^-)_{k\in\mathbb{Z}}$ decompose $\partial_t^sf^p$.
%
%\begin{eqnarray}
%{\beta}_{k}^{+}(f) &=& \frac{p}{2} \Psi_k^+(f) \nonumber \\
%{\beta}_k^-(f) &=& \frac{p}{2} \Psi_k^-(f) \nonumber \\
%\end{eqnarray}
%
$({\theta}_{k}^{+})_{k\in\mathbb{Z}}$ and $({\theta}_{k}^{-})_{k\in\mathbb{Z}}$ are proportional to  $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$  and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ respectively. Finally, $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$  and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$  decompose the successive derivatives of ${f}^{p}$ ($p \in \mathbb{Z}^+$, and $p> 1$).
%
\vspace{1.5em}
%
\newline B- $\bold{Existence}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$ 
%
\vspace{1.5em}
%
\newline The proof is also structured as an induction on the $n$-th power of $f$. The different cases revisit some families of operator defined in the previous sections of this work (e.g., proof of the Lemma).
%for the case $n=\{2,3,p\}$ in order to support the existence of the decomposition. 
%
\vspace{1.5em}
%
\\$\bold{Case}$ $n=2$: 
%
%\vspace{1.5em}
%
It was shown that the family of operator $(\eta_k)_{k\in\mathbb{Z}}$ (see definition in the proof of the Lemma), decomposes $\partial^s_t f^{2}$ ($s\in\mathbb{Z}^+-\{0\}$). As defined in Equation \eqref{definitionOfeta}, one can rewrite it as a sum of the DEO family $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ as:
%
\begin{equation}
\eta_k(f) = \Psi_k^+(f) + 2 \Psi_k^-(f), \qquad k \in \mathbb{Z}
\end {equation}
%
%
\vspace{1.5em}
%
\\$\bold{Case}$ $n=3$: 
%
%\vspace{1.5em}
%
In the lemma, the family of operator $(\Gamma_{k}^{+})_{k\in\mathbb{Z}}$ was defined in Equation \eqref{GammaOperatorfamily} and decomposes $\partial^s_t f^{3}$ ($s\in\mathbb{Z}^+-\{0\}$). One can rewrite the definition as:
%
%
\begin{eqnarray}\label{GammaOperatorfamilyPi}
%\Gamma_{k}^{+} (f) = 2 \dot{f_t}{f_t}^{k-1} + f_t {f_t}^{k} \qquad k=\{0,\pm 1,\pm 2,...\}
%
\Gamma_{k}^{+} (f) &=& \frac{3}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\
\Gamma_{k}^{+} (f) &=& \frac{3}{2} \Psi_k^+(f) + 0 \Psi_k^-(f)
%
\end{eqnarray}
%
\vspace{1.5em}
%
\\$\bold{Case}$ $n=p$ with $p>1$: 
%
%\vspace{1.5em}
%
Previously, Equations \eqref{psi+0008} defined the operators ${\theta}_{k}^{+}$ and ${\theta}_{k}^{-}$ decomposing $\partial^{s}_t f^{p}$ ($s\in\mathbb{Z}^+-\{0\}$) and proportional to $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ as:
%
%\begin{eqnarray}\label{psi+000100}
\begin{eqnarray}\label{psi+000100}
{\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\
&= & \frac{(p-1)}{2} \Psi_k^+(f) \nonumber \\
{\theta}_{k}^{-}(f) &= & \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} - f {f}^{(k)}) \nonumber \\
&= &\frac{(p-1)}{2} \Psi_k^-(f)
\end{eqnarray}
%
By induction it was also shown in the same section that ${\theta}_{k}^{-}$ and ${\theta}_{k}^{+}$ decompose $\partial^{s}_t f^{p+1}$ ($s\in\mathbb{Z}^+-\{0\}$). It is then possible to conclude the existence of the decomposition of any operator by using $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$.
%
%Using the derivative chain rule property, one can define the families of operator $(\psi_{k}^{+})_{k\in\mathbb{Z}}$ and $(\psi_{k}^{-})_{k\in\mathbb{Z}}$ as:
%%
%
%\begin{eqnarray}
%
%\\ Note that based on the similar development done in Equations (\ref{Coeff2nmoins}), one can show that the DEO family $\Psi_{k}^{-}$ ($k \in \mathbb{Z}$) decomposes also $\Psi^{-}_k$ for any function in $Ker(\Psi^{-}_k)$ ($k$ in $\mathbb{Z}$).
%
\vspace{1.5em}
%
\newline C-  $\bold{About}$ $\bold{Uniqueness}$ $\bold{of}$ $\bold{the}$  $\bold{Decomposition}$ 
%
\vspace{1.5em}
%
%%
%%Then, $Im(\Psi^{-}_1)$ and $Im(\Psi^{+}_1)$ are in direct sum. 
%Following the definition of $Im(\partial_t^p f^n)$ ( $p\in\mathbb{Z}^+-\{0\}$, $n\in\mathbb{Z}^+$, and $n>1$):
%%
%\begin{equation}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
%Im(\partial_t^p f^n) = \{\partial_t^p f^n \in \mathbb{R} | f \in \mathbf{S}^{-}(\mathbb{R}) \}
%\end{equation} 
%%
%%Then by definition of the direct sum,
%Based on the existence of the decomposition demonstrated in the first paragraph, one can then write:
%%
%\begin{equation}
%Im(\partial_t^{p-1}\Psi^{+}_1) + Im(\partial_t^{p-1}\Psi^{-}_1) = Im(\partial_t^p f^n), \qquad p\in\mathbb{Z}^+-\{0\}, \qquad n\in\mathbb{Z}^+ -\{0,1\}
%\end{equation}
%%
\\ With the previous section, let us show by induction the uniqueness of the decomposition of any family of operators decomposing $\partial_t^s f^n$ ($s$ in $\mathbb{Z}^+$, $n$ in $\mathbb{Z}^+$ and $n > 1$). The induction is focused on the proof of the uniqueness of the decomposition of a family of operator $({S}_{k})_{k\in\mathbb{Z}}$ (following the derivative chain rule property) by $({\Psi}_{k}^+)_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^-)_{k\in\mathbb{Z}}$. In other words, the induction is on the $k$-th order of the operator. 
%
%
%\vspace{1.5em}
%%
%\\$\bold{Case}$ $n=2$: 
%%
%\vspace{1.5em}
%%
\vspace{1.5em}
%
\\$\bold{Case}$ $k=2$: 
%
For $f$ in $\mathbf{s}^{-}(\mathbb{R})$ and $n$ in $\mathbb{Z}^+$ and $n > 1$, one can assume that $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ exist in $\mathbb{R}^4$ such as:
%%
%
%
%
%Let us write the 
%
\begin{eqnarray}\label{eqrefpartial0102}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
\partial_t^s f^n &=& \partial_t^{s-1} (f^{n-2} S_1(f))  \nonumber \\
\partial_t^s f^n &=&  \partial_t^{s-1} ( f^{n-2} (\alpha_1 \Psi^{+}_1(f) +\alpha_2 \Psi^{-}_1(f))) \nonumber \\
\partial_t^s f^n &=& \partial_t^{s-1} ( f^{n-2} (\beta_1 \Psi^{+}_1(f) +\beta_2 \Psi^{-}_1(f))) \nonumber \\
\end{eqnarray} 
%% 
%
%\vspace{1.5em}
%
\\ As  with the operator family $({S}_{k})_{k\in\mathbb{Z}}$ follows the derivative chain rule property:
%
% If we write the above equation for $p=1$ and differentiate them, this leads to:
%%
\begin{eqnarray}\label{equationalphabeta1}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
\partial_t S_1(f) &=& S_2(f) +S_0(\partial_t f)\nonumber \\
\partial_t S_1(f) &=&\alpha_1 \partial_t \Psi^{+}_1(f) +\alpha_2 \partial_t \Psi^{-}_1(f) \nonumber \\
\partial_t S_1(f) &=&\alpha_1 (\Psi^{+}_2(f) +\Psi^{+}_0(\partial_t f)) +\alpha_2 ( \Psi^{-}_2(f) +\Psi^{-}_0(\partial_t f)) \nonumber \\
\end{eqnarray} 
%
And then,
%
\begin{eqnarray}\label{equationalphabeta1b}
S_2(f) &=& \alpha_1 \Psi^{+}_2(f) +\alpha_2 \Psi^{-}_2(f) \nonumber \\
S_2(f) &=& \beta_1 \Psi^{+}_2(f) +\beta_2 \Psi^{-}_2(f) \nonumber \\
(\alpha_1-\beta_1) \Psi^{+}_2(f) +(\alpha_2-\beta_2) \Psi^{-}_2(f) &=& 0
%\partial_t^p f^2 &=& \beta_1 \partial_t \Psi^{+}_1(f) +\beta_2 \partial_t \Psi^{-}_1(f) \nonumber \\
%(\alpha_1-\beta_1)\partial_t \Psi^{+}_1(f) + (\alpha_2-\beta_2)\partial_t \Psi^{-}_1(f) &=& 0 \nonumber \\
\end{eqnarray} 
%%
As $Im(\Psi^{+}_2)$ and $Im(\Psi^{-}_2)$ are not reduced to $\{0\}$ by definition, it follows that $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$. Note that it is not possible to do this simple check for $k=1$ as $Im(\Psi^{-}_1)=\{0\}$.
%
\vspace{1.5em}
%
\\$\bold{Case}$ $k=p$: 
%
Let us assume the uniqueness of the decomposition for $k=p-1$ (with $k\neq1$). For $k=p$, following Equation \eqref{equationalphabeta1}:
%
\begin{eqnarray}\label{equationalphabeta1kn}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
\partial_t S_{p-1}(f) &=& S_p(f) +S_{p-2}(\partial_t f)\nonumber \\
\partial_t S_{p-1}(f) &=&\alpha_1 \partial_t \Psi^{+}_{p-1}(f) +\alpha_2 \partial_t \Psi^{-}_{p-1}(f) \nonumber \\
\partial_t S_{p-1}(f) &=&\alpha_1 (\Psi^{+}_p(f) +\Psi^{+}_{p-2}(\partial_t f)) +\alpha_2 ( \Psi^{-}_p(f) +\Psi^{+}_{p-2}(\partial_t f)) \nonumber \\
\end{eqnarray} 
%
And then,
%
%
\begin{eqnarray}\label{equationalphabeta1knb}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
S_p(f) &=& \alpha_1 \Psi^{+}_p(f) +\alpha_2 \Psi^{-}_p(f) \nonumber \\
S_p(f) &=& \beta_1 \Psi^{+}_p(f) +\beta_2 \Psi^{-}_p(f) \nonumber \\
(\alpha_1-\beta_1) \Psi^{+}_p(f) +(\alpha_2-\beta_2) \Psi^{-}_p(f) &=& 0
%\partial_t^p f^2 &=& \beta_1 \partial_t \Psi^{+}_1(f) +\beta_2 \partial_t \Psi^{-}_1(f) \nonumber \\
%(\alpha_1-\beta_1)\partial_t \Psi^{+}_1(f) + (\alpha_2-\beta_2)\partial_t \Psi^{-}_1(f) &=& 0 \nonumber \\
\end{eqnarray} 
%
By definition for $p\neq1$, $Im(\Psi^{+}_p)$ and $Im(\Psi^{-}_p)$ are not reduced to $\{0\}$, and it follows that $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$. 
%
\vspace{1.5em}
%
\\$\bold{Special}$ $\bold{Case}$ $k=1$: 
%
To complete the proof with the assumption that  $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$ for $k \in\mathbb{Z}$ and $k \neq 1$ , the special case $k=1$ can be solved as:
% 
\begin{eqnarray}\label{equationalphabeta1c}
\partial_t(\alpha_1 \Psi^{+}_1(f)) &=&\alpha_1(\Psi^{+}_2(f) + \Psi^{+}_0(\partial_t f)) \nonumber \\
&=&\beta_1(\Psi^{+}_2(f) + \Psi^{+}_0(\partial_t f)) \nonumber \\
&=&\partial_t(\beta_1\Psi^{+}_1(f))\nonumber \\
& & \nonumber \\
\partial_t(\alpha_2 \Psi^{-}_1(f)) &=&\alpha_2(\Psi^{-}_2(f) + \Psi^{-}_0(\partial_t f)) \nonumber \\
&=&\beta_2(\Psi^{-}_2(f) + \Psi^{-}_0(\partial_t f)) \nonumber \\
&=&\partial_t(\beta_2\Psi^{-}_1(f))\nonumber \\
\nonumber\\
\end{eqnarray} 
%
Thus, $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$. Let us finish the proof with a remark for the case $\partial_t f^2$. Rewriting Equation \eqref{eqrefpartial0102} as:
%
\begin{eqnarray}
%%\mathbb{S} = \{f\in C^{\infty}| , \qquad f(t)=0, \qquad \forall t\in \mathbb{R}\}
\partial_t f^2 &=& S_1(f)  \nonumber \\
\partial_t f^2 &=& \alpha_1 \Psi^{+}_1(f) +\alpha_2 \Psi^{-}_1(f) \nonumber \\               
\partial_t f^2 &=& \beta_1 \Psi^{+}_1(f) +\beta_2  \Psi^{-}_1(f) \nonumber \\
&=&  \Psi^{+}_1(f) 
\end{eqnarray} 
%
Thus, we conclude that in this case, $\alpha_1 = \beta_1 =1$.
%With this equation, one can only conclude that for any  $f$ in $\mathbf{S}^{-}(\mathbb{R}) \}$ $\alpha_1=\beta_1$. Before continuing the development, let remind us the Equations \eqref{Coeff2nmoins} where we stated for any $f$ in $\mathbf{S}^{-}(\mathbb{R}) \}$, ${\Psi}_{2}^{-}(f) = -{\Psi}_{0}^{-}(\partial_t f)$. In the same way, based on the definition of $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$: 
%%
% %{\Gamma}_{2}^{-}(f) - {\Psi}_{2}^{-}(f) &=&  {\Psi}_{0}^{-}(\partial f) -{\Gamma}_{0}^{-}(\partial f) \nonumber \\
% ({\Gamma}_{p+1}^{-}(f) - {\Psi}_{p+1}^{-}(f)) + (-{\Psi}_{-p+1}^{-}(\partial_t^p f) +{\Gamma}_{-p+1}^{-}(\partial_t^p f)) &=& 0 \nonumber \\
% {\Gamma}_{p+1}^{-}(f) &=& {\Psi}_{p+1}^{-}(f) \nonumber \\
% {\Psi}_{-p+1}^{-}(\partial_t^p f)  &=&{\Gamma}_{-p+1}^{-}(\partial_t^p f)
% \end{eqnarray}
% %
% This gives the uniqueness of the decomposition.
%
\end{proof}
%
Note that in \cite{Boudraa et al.2009}, the authors based their work on the energy operator defined as $LP_2(f,g)=\partial_t f \partial_t g - f \partial_t^2 g$ ($LP_2$ : $\mathbb{R}^2$ $\rightarrow$ $\mathbb{R}$) which was then generalized to the complex set ($LP_2^{\mathbb{C}}(f,g)$). In the development of their work, they found a similar type of formula as found in Equation \eqref{coefficientsAPgeneral}, but restricted to the definition of their energy operator. In addition, one can underline that the generalization of the decomposition of the successive derivatives of the $n$-th power of $f$ with the DEOs ${(\Psi^{+}_k)}_{k\in \mathbb{Z}}$ and ${(\Psi^{-}_k)}_{k\in \mathbb{Z}}$ (e.g., Equations \eqref{coefficientsAPgeneral} and \eqref{Coeff2nmoinspp}) follows the general Leibniz derivative rules \cite{BruceWest}. 
% 
%\section{Examples - Schroedinger Equation - KdV Equation - Hamiltonian}
%
% General Decomposition
%
%\section{Discussion for $n=1$ and $n<0$}
%
\vspace{0.5em}
%
\newline $\bold{Discussion}$ $n<-1$: This case focuses on the decomposition using the DEO family of the quotient of the function $f$:($\mathbb{R}\rightarrow\mathbb{R}$) defined as:
%
\begin{equation}
\forall f \in \mathbf{S}^{-}(\mathbb{R}), \qquad \forall t \in \mathbb{R}, \qquad f(t)\neq 0, \qquad \forall n \in \mathbb{Z}^+, n>1, \frac{1}{f^n}
%\{\frac{1}{f^n}, \frac{1}{f^3}, \frac{1}{f^4}, ...\}
\end{equation}
%
They are just a particular case of $f^n$ ($n>1$, $n\in \mathbb{Z}^+$). Using an intermediary function, $h$ such as $h = \frac{1}{f}$, the problem of decomposing $\partial_t^s f^{-n}$ ($s \in \mathbb{Z}^+$) is equivalent to resolving $\partial_t^s h^{n}$, which has been demonstrated in the Lemma and Theorem.
%
\vspace{1.5em}
%
\newline $\bold{Discussion}$ $n=1$: This case does not make sense to decompose $f$ and its $k$-th derivative ($k\in\mathbb{Z}^+$) as a sum of energy operators based on the general definition of the DEO from \cite{Maragos1995}: the $k$-th order DEO is the cross energy between a function and its $k-1$ derivatives. However, one can use a general formula in the special case:
%
\begin{equation}
\forall t\in \mathbb{R} , \qquad f\in \mathbf{S}^{-}(\mathbb{R}),\qquad f(t) \neq 0 \nonumber
\end{equation}
%
\begin{eqnarray}\label{discussion2a}
\partial_t^k f &=& \partial_t^k \big(\frac{f^3}{f^2}\big) \nonumber \\
k=1, \qquad \partial_t f &=& f^{-2} \partial_t f^3 + f^3 \partial_t f^{-2} \nonumber \\
k=2, \qquad \partial_t^2 f &=& 2 \partial_t f^{-2} \partial_t f^3 + f^3 \partial_t^2 f^{-2} + f^{-2} \partial_t^2 f^3
\end{eqnarray} 
%
The example for $k =\{1,2\}$ in Equation \eqref{discussion2a} shows that $\partial_t f$  can be decomposed into a product of successive derivatives of $f^{3}$ and $f^{-2}$. Those derivatives can be decomposed into a sum of DEOs based on the development of the proof of the Lemma and the previous discussion (for $n<-1$ ).
%
\vspace{0.5em}
%
% Look at the representation between Sigma_+ and Sigma_- ...
%
%
\newline Note that if we restrict $f$ in $\mathbf{S}^{-}(\mathbb{R})$ and to be expandable in Taylor-Series in an interval $[a,b]$ in $\mathbb{R}$ such as:
%
\begin{equation}
f(t) = \Sigma_{k=0}^{\infty} (\partial_t^k f(t_0)) \frac{(t-t_0)^k}{k!}, \qquad \forall t \in [a,b], \qquad t_0 \in [a,b] 
\end{equation}
%
One can then decompose the coefficients of the Taylor-Series of $f$ as a sum of DEO via the method shown in the Lemma and Theorem.
%theorem is developed using the Taylor-Mc Laurin series, but as $f$ is in $C^{\infty}$ (f is continuous and differentiable for all $t$ in $\mathbb{R}$), the general formula of the Taylor series can be used instead.    
%The Theorem is the well-posed problem of defining a function to link a LPDE to the Energy domain
%
%\vspace{1.5em}
%
%\newline One can state a corollary of Lemma $2$. Let us define the set $\mathbf{E}$ such as:
%
%\begin{equation}
%\mathbf{E} =\{f \in \mathbf{S}^{-}(\mathbb{R}) |  \qquad \Psi_{k}^{\pm}(f) \in \mathbb{R}, \qquad k \in \mathbb{Z}\} 
%\end{equation}
%
%$\Psi_{k}^{\pm}$ refers to $\Psi_{k}^{-}$ and $\Psi_{k}^{+}$. It is then possible to define:
%
%\begin{equation}
%Ker(\mathbf{E}) =\{f \in \mathbf{S}^{-}(\mathbb{R})|  \qquad \Psi_{k}^{\pm}(f)=0, \qquad k \in \mathbb{Z}\} 
%\end{equation}
%
%Note $Ker(\mathbf{E})$ is non empty as the null function is included.
%
%\vspace{1.5em}
%
%\\$\mathbf{Corollary}$ (to Lemma $2$): the DEO family $\Psi_{k}^{-}$ (k in $\mathbb{Z}$) decomposes ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ for any function $f$ in $Ker(\mathbf{E})$.
%
\section{Some Properties of the Images and Kernels of the DEO Families}\label{SectionProperties}
%
In this part, we study the relationships between $Im(\Psi^{+}_k)$ and $Im(\Psi^{-}_k)$, and $Ker(\Psi^{+}_k)$ and $Ker(\Psi^{-}_k)$  via the demonstration of the following properties.
%
\\$\mathbf{Properties}$ $1$: for $k$ in $\mathbb{Z}$ and $f$ in $\mathbf{S}^{-}(\mathbb{R})$,
\begin{eqnarray}\label{Property01}
a)  \Psi^{+}_k(f) &=& \Psi^{+}_{-k+2}(\partial_t^{k-1}f) \nonumber \\
b) \Psi^{-}_k(f) &=& -\Psi^{-}_{-k+2}(\partial_t^{k-1}f) \nonumber \\
c) \Psi^{+}_k(f)+\Psi^{-}_k(f) &=& \Psi^{+}_{-k+2}(\partial_t^{k-1}f) -\Psi^{-}_{-k+2}(\partial_t^{k-1}f) \nonumber \\ 
d) Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) & \subseteq & Im(\Psi^{+}_k - \Psi^{-}_k) \nonumber \\
e) Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) & \subseteq & Im(\Psi^{+}_k + \Psi^{-}_k) \nonumber \\
%f) Im(-\Psi^{+}_k) &=& -Im(\Psi^{+}_k) \nonumber \\
%g) Im(-\Psi^{-}_k) &=& -Im(\Psi^{-}_k) \nonumber \\ 
\end{eqnarray}
%
\begin{proof}
To show Equation (\ref{Property01}-a), the definition of $\Psi^{+}_{-k+2}(\partial_t^{k-1}f)$ is:
%
\begin{equation}
\Psi^{+}_{-k+2}(f) = \partial_t f \partial_t^{-k+1}f + f \partial_t^{-k+2}f
\end{equation}
%
and then,
%
\begin{eqnarray}\label{equationproperty01b}
\Psi^{+}_{-k+2}(\partial_t^{k-1}f) &=& \partial_t^k f f + \partial_t^{k-1}f \partial_t f \nonumber \\
\Psi^{+}_{-k+2}(\partial_t^{k-1}f) &=& \Psi^{+}_k(f) 
\end{eqnarray}
%
This last equation then shows the first assertion $\Psi^{+}_k(f) = \Psi^{+}_{-k+2}(\partial_t^{k-1}f)$. Following the same development as in Equation \eqref{equationproperty01b}, one can derive directly Equation (\ref{Property01}-b) as:
%
\begin{eqnarray}\label{equationproperty01c}
\Psi^{-}_{-k+2}(\partial_t^{k-1}f) &=& \partial_t^k f f - \partial_t^{k-1}f \partial_t f \nonumber \\
\Psi^{-}_{-k+2}(\partial_t^{k-1}f) &=& -\Psi^{-}_k(f) 
\end{eqnarray}
%
Equation (\ref{Property01}-c) is a consequence of  Equation (\ref{Property01}-a) and Equation (\ref{Property01}-b) by simply replacing  $\Psi^{+}_k(f)$ and $\Psi^{-}_k(f)$ by respectively $\Psi^{+}_{-k+2}(\partial_t^{k-1}f)$ and $-\Psi^{-}_{-k+2}(\partial_t^{k-1}f)$. 
%
\newline Let us write the definition of $Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) $ and $Im(\Psi^{+}_k - \Psi^{-}_k)$ such as:
%
\begin{eqnarray}
 Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) &=& \{\Psi^{+}_k(f) = \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\ 
                                                             &=& \{ 2 f\partial_t^k f =0 |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
Im(\Psi^{+}_k - \Psi^{-}_k) &=& \{\Psi^{+}_k(f) - \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
                                           &=& \{ 2 f\partial_t^k f |f \in \mathbf{S}^{-}(\mathbb{R})\} 
\end{eqnarray}
%
$Im(\Psi^{+}_k - \Psi^{-}_k)$ is non-empty as it contains $0$ when $f$ is the null function. Thus, the above definitions show the inclusion of the subset $Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k)$ into $Im(\Psi^{+}_k - \Psi^{-}_k)$. In the same way, one can show Equation (\ref{Property01}-e) through the definition of each subset.
%
\begin{eqnarray}
 Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) &=& \{\Psi^{+}_k(f) +\Psi^{-}_k(f) =0 |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\ 
Im(\Psi^{+}_k + \Psi^{-}_k) &=& \{\Psi^{+}_k(f) + \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\                              
\end{eqnarray}
%
As before, $0$ is included in $Im(\Psi^{+}_k + \Psi^{-}_k)$ (e.g. $\Psi^{+}_k (0) = \Psi^{-}_k (0) =0$, for $k$ in $\mathbb{Z}$). Thus, $Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) \subset Im(\Psi^{+}_k + \Psi^{-}_k)$.
%
Note that Equations (\ref{Property01}-a), (\ref{Property01}-b) and (\ref{Property01}-c) are important as they are directly linked to $Im(\partial_t^p\Psi^{+}_1)$ and $Im(\partial_t^p\Psi^{-}_1)$ ($p$ in $\mathbb{Z}-\{0\}$).% and also to the formulas given in Equations \eqref{Coeff2n}, \eqref{Coeff2nmoins} and \eqref{Coeff2nmoinspp}.
%
%\newline  Equations (\ref{Property01}-e) and Equations (\ref{Property01}-f) are also directly demonstrated by the definition of  $Im(\Psi^{+}_k)$ and $Im(\Psi^{-}_k)$ given in Equation \eqref{Impsik+}.
%
\end{proof}
%
Similar properties can be shown from the definitions of $Ker(\Psi^{+}_k)$ and $Ker(\Psi^{-}_k)$:
%
\\$\mathbf{Properties}$ $2$: for $k$ in $\mathbb{Z}$ and $f$ in $\mathbf{S}^{-}(\mathbb{R})$,
\begin{eqnarray}\label{Property02Kernel}
%a)  Ker(\Psi^{+}_k(f)) &=& Ker(\Psi^{+}_{-k+2}(\partial_t^{k-1}f)) \nonumber \\
%b) Ker(\Psi^{-}_k(f)) &=& -Ker(\Psi^{-}_{-k+2}(\partial_t^{k-1}f)) \nonumber \\
%c) Ker(\Psi^{+}_k(f)+\Psi^{-}_k(f)) &=& Ker(\Psi^{+}_{-k+2}(\partial_t^{k-1}f) -\Psi^{-}_{-k+2}(\partial_t^{k-1}f)) \nonumber \\ 
a) Ker(\Psi^{+}_k) \bigcap Ker(\Psi^{-}_k) & \subseteq & Ker(\Psi^{+}_k - \Psi^{-}_k) \nonumber \\
b) Ker(\Psi^{+}_k) \bigcap Ker(-\Psi^{-}_k) & \subseteq & Ker(\Psi^{+}_k + \Psi^{-}_k) \nonumber \\
c) Ker(-\Psi^{+}_k) &=& Ker(\Psi^{+}_k) \nonumber \\
d) Ker(-\Psi^{-}_k) &=& Ker(\Psi^{-}_k) \nonumber \\ 
\end{eqnarray}
%\section{Extending the results from time to ($3$D) Space}
%
\begin{proof}
%%->
%Equations (\ref{Property02Kernel}-a), (\ref{Property02Kernel}-b) and (\ref{Property02Kernel}-c) are based on the same remark as done in Equations (\ref{Property01}-a), (\ref{Property01}-b) and (\ref{Property01}-c) such as for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$:
%
%\begin{eqnarray}\label{KernelExpress01}
%\Psi^{+}_k(f) = \Psi^{+}_{-k+2}(\partial_t^{k-1}f) \nonumber \\
%\Psi^{-}_k(f) = -\Psi^{-}_{-k+2}(\partial_t^{k-1}f) \nonumber \\
%\end{eqnarray} 
%
%Reminding the definition of $Ker(\Psi^{+}_k(f))$, it follows:
%
%\begin{eqnarray}
%Ker(\Psi^{+}_k (f)) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})|  \qquad \Psi^{+}_k(f)=0 \} \nonumber \\
%Ker(\Psi^{+}_k (f)) &=& Ker(\Psi^{+}_{-k+2}(\partial_t^{k-1}f)) \nonumber
%\end{eqnarray}
%
%And similarly, 
%
%\begin{eqnarray}
%Ker(\Psi^{-}_k (f)) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})|  \qquad \Psi^{-}_k(f)=0 \} \nonumber \\
%Ker(\Psi^{-}_k (f)) &=& Ker(\Psi^{-}_{-k+2}(\partial_t^{k-1}f)) \nonumber
%\end{eqnarray}
%
%Equation (\ref{Property02Kernel}-c) is directly shown by literally replacing  $\Psi^{+}_k (f)$ and $\Psi^{-}_k (f)$ with the expression in Equation \eqref{KernelExpress01}.
%
The demonstration follows the definition of the kernels in the same way as in the previous properties:
%
\begin{eqnarray} \label{defKer01}
Ker(\Psi^{+}_k) \bigcap Ker(\Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \qquad \Psi^{-}_k(f)=\Psi^{+}_k(f) =0\} \nonumber \\
%
\end{eqnarray}
%
\begin{eqnarray}\label{defKer02}
Ker(\Psi^{+}_k - \Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) -\Psi^{-}_k(f) =0\} \nonumber \\
\end{eqnarray}
%
And, 
\begin{eqnarray} \label{defKer01b}
Ker(\Psi^{+}_k) \bigcap Ker(-\Psi^{-}_k)    &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \qquad \Psi^{+}_k(f)=-\Psi^{-}_k(f) =0\} \nonumber\\
\end{eqnarray}
%
%
\begin{eqnarray}\label{defKer02b}
%Ker(\Psi^{+}_k - \Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) -\Psi^{-}_k(f) =0\} \nonumber \\
Ker(\Psi^{+}_k + \Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) +\Psi^{-}_k(f) =0\}\nonumber \\
\end{eqnarray}
%
It is necessary to underline that $Ker(\Psi^{+}_k)$, $Ker(\Psi^{-}_k)$ , $Ker(\Psi^{+}_k + \Psi^{-}_k)$ and $Ker(\Psi^{+}_k - \Psi^{-}_k)$ are all non empty sets as the null function is included in all of them.  This then demonstrates Equations (\ref{Property02Kernel}-a) and (\ref{Property02Kernel}-b).
%The definitions in Equations \eqref{defKer01} and \eqref{defKer02} justify properties (\ref{Property02Kernel}-d) and (\ref{Property02Kernel}-e).
%
\newline Similarly we have,
%
\begin{eqnarray}\label{defKer03}
Ker(\Psi^{+}_k ) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) = -\Psi^{+}_k(f) =0\} \nonumber \\
Ker(\Psi^{-}_k ) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{-}_k(f) = -\Psi^{-}_k(f) =0\} \nonumber \\
\end{eqnarray}
%
This directly shows Equations  (\ref{Property02Kernel}-c) and (\ref{Property02Kernel}-d). 
%
%According to \cite{JPMontillet2010}, the notion of energy operator in space (${\Psi}_{R}^{s+}$, ${\Psi}_{R}^{s-}$) was introduced as part of the method developed to decompose the wave equation. Let us define a function $\alpha$ ($\mathbb{R}^3$ $\rightarrow$ $\mathbb{R}$ in $C^{\infty}(f)$) in $\bold{space}$ ($3$D). Note in the notation of ${\Psi}_{R}^{s+}$, $s$ stands for space. Here the development is given in one dimensional case such as: 
%%
\end{proof}
%
Following the remarks at the end of Properties $1$, we can rewrite the formulas in Equations \eqref{coefficientsAPgeneral} based on the relationship of the images. With the Equation (\ref{Property01}-a), the formula \eqref{coefficientsAPgeneral} can be rewritten as:
%
\begin{eqnarray}\label{Coeff2nRewritten}
%
\partial_t {\Psi}_{1}^{+}(f) &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) \nonumber \\
\partial_t {\Psi}_{1}^{+}(f) &=& 2{\Psi}_{2}^{+}(f) \nonumber \\
%
\partial_t^2 {\Psi}_{1}^{+}(f) &=& {\Psi}_{3}^{+}(f) + 2{\Psi}_{1}^{+}(\partial_t f) + {\Psi}_{-1}^{+}(\partial_t^2 f) \nonumber \\
&=& 2{\Psi}_{3}^{+}(f) + 2{\Psi}_{1}^{+}(\partial_t f) \nonumber \\
%
\end{eqnarray}
%
%Let us define the function \emph{rounding} $g_r: \mathbb{R} \rightarrow \mathbb{Z}$ such as:
%
%
Thus for $p$ in $\mathbb{Z}^{+}-\{0\}$, $k$ in $\mathbb{Z}$ and $s$ in in $\mathbb{Z}^+-\{0\}$,
%
\begin{equation*}\label{coefficientsAPgeneralRewrittenA}
%a_p^+(f) &=& \sum_{k=0}^{p-1} \big(_{k}^{p-1} \big) \Psi_{2(k+1)-p}^{+}(\partial_t^{p-k-1}f) \nonumber \\
a_p^+(f) =\left\{
 \begin{array}{rl}
 \Psi_{1}^{+}(f),&  p=1 \nonumber \\
2 \sum_{k > \frac{2s-1}{2}}^{2s-1} \big(_{k}^{2s-1} \big)\Psi_{2(k+1)-2s}^{+}(\partial_t^{2s-k-1}f), &  p=2s  \\
2 \sum_{k > s}^{2s} \big(_{k}^{2s} \big) \Psi_{2(k+1)-2s-1}^{+}(\partial_t^{2s-k}f) + \big(_{s}^{2s} \big) \Psi_{1}^{+}(\partial_t^{s}f),&  p=2s+1 \\
\end{array} \right .
\end{equation*}
%
Consequently, the successive derivatives of $f^n$ ($n$ in $\mathbb{Z}^{+}-\{0\}$) can be decomposed with the DEO family $(\Psi_{k}^{+})$ with $k$ in $\mathbb{Z}^+-\{0\}$. 
%
\newline In the same way following Equation (\ref{Property01}-b), Equation \eqref{coefficientsAPgeneral2bis} is simplified to:
%
\begin{equation}\label{coefficientsAPgeneralRewrittenA}
%a_p^+(f) &=& \sum_{k=0}^{p-1} \big(_{k}^{p-1} \big) \Psi_{2(k+1)-p}^{+}(\partial_t^{p-k-1}f) \nonumber \\
a_p^-(f) = 0 \qquad \forall  p \in \mathbb{Z}^{+}-\{0\}
\end{equation}
%
%
%\begin{eqnarray}\label{Energy operator in space in one dimension}
%{\Psi}_{R}^{s \{ \partial x \}-}[\alpha] &=& \frac{ \partial \alpha(x)}{\partial x}\frac{d\alpha(x)}{\partial x}- \frac{\partial^2\alpha(x)}{\partial x^2}\alpha(x) \nonumber \\
%{\Psi}_{R}^{s \{ \partial x \}+}[\alpha] &=& \frac{\partial \alpha(x)}{\partial x}\frac{d\alpha(x)}{\partial x}+ \frac{\partial^2\alpha(x)}{\partial x^2}\alpha(x) \nonumber \\
%\end{eqnarray}   
%%
%with ${\Psi}_{R}^{s \{ \partial x \}-}$, the energy operator in space developed following the $x$ Cartesian coordinate ($\partial x$ in the notation). Following the definition in time of the energy operator given in Equation \eqref{familyDEOdefinition}, this can be extended to space by switching the variable :
%%
%\begin{eqnarray}\label{DEO family in space}
%{\Psi}_{k}^{s \{ \partial x \}+}(\alpha) = \partial_x \alpha \partial_x^{(k-1)} \alpha + \alpha \partial_x^{(k)}\alpha, \qquad k \in \mathbb{Z} \nonumber \\
%{\Psi}_{k}^{s \{ \partial x \}-}(\alpha) = \partial_x \alpha \partial_x^{(k-1)} \alpha - \alpha \partial_x^{(k)}\alpha, \qquad k \in \mathbb{Z} 
%\end{eqnarray}
%%
%By switching the variable from time to space, the properties of the DEO remain the same in space (e.g. bilinearity, differentiation rules). The whole development of the lemmas and the theorem in the previous section is then true for the family of DEO in space as described in Equations \eqref{DEO family in space}. Note that the extension from one dimension to $3$ is easily done by adding the energy operators (${\Psi}_{k}^{s \{ \partial x \}+} + {\Psi}_{k}^{s \{ \partial y \}+} + {\Psi}_{k}^{s \{ \partial z \}+}$ ) as it was previously described in \cite{JPMontillet2010}.
%% 
%
\section{Application to the Energy Function}
%
The energy function $\mathcal{E}$ considered in this section is the one previously defined in Equation \eqref{EnergyfunDefine02}.
%
$f_1$ and $f_1^2$ are considered to be in $\mathbf{S}^{-}(\mathbb{R})$, analytic and with a finite energy. 
%
Note that the choice of this example is also based on the discussion (case $n=1$) after the Lemma. %The first part is the development of $\mathcal{E}$ with the DEO families $({\Psi}_{k}^{+})_{k \in \mathbb{Z}}$ and $({\Psi}_{k}^{-})_{k \in \mathbb{Z}}$. %Then, the second part is the relationship with the Dirichelet-Parseval formula.
%
%\subsection{Energy function and DEO family}
%
\\ One can develop the energy function in Taylor-Series on the interval of definition $[0,\tau]$ ($\tau$ in $\mathbb{R}$) for a nominated $\tau_0$ in the defined interval,
%
\begin{eqnarray}\label{EquationEfunction}
\mathcal{E}(f_1(\tau) ) &=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^k \mathcal{E}(f_1(\tau_0)) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
                                 &=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^{k-1} f_1^2(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
                                  &=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^{k-1} f_1^2(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
                                  &=& \mathcal{E}(f_1(\tau_0)) +f_1^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} (\Psi_1^+(f_1)(\tau_0)   \nonumber \\
& &+ \Psi_1^-(f_1)(\tau_0))\frac{(\tau-\tau_0)^k}{k!} \nonumber \\
&=& \mathcal{E}(f_1(\tau_0)) +f_1^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} \Psi_1^+(f_1)(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
\end{eqnarray}
%
In the special case that the series is absolutely convergent, one can write for $p$ in $\mathbb{Z}^+-\{0\}$:
%
\begin{eqnarray}
\big|  \frac{\partial_t^{p+1} \mathcal{E}(f_1(\tau_0)) }{\partial_t^{p} \mathcal{E}(f_1(\tau_0))} \big| \big| \frac{(\tau-\tau_0)}{p+1} \big|&<& 1 \nonumber \\
\end{eqnarray}
%
or for  $p$ in $\mathbb{Z}^+-\{0,1\}$
%
\begin{eqnarray}\label{equationenergyseries1}
\big| \frac{\partial_t^{p+1} (\Psi_1^+(f_1)(\tau_0) }{\partial_t^{p} \Psi_1^+(f_1)(\tau_0) } \big| &<& \big| \frac{p+1}{(\tau-\tau_0)}\big|
\end{eqnarray}
%
%if $\big| \frac{p+1}{(\tau-\tau_0)} \big| =\alpha_p(\tau_0) $ which is positive, then:
%
%\begin{eqnarray}
%| \partial_t^{p+1} \Psi_1^+(f_1)(\tau_0) |&<& \alpha_p(\tau_0) | \partial_t^{p} \Psi_1^+(f_1)(\tau_0) | \nonumber \\
%&<& \alpha_p(\tau_0) | \partial_t^{p-1} f_1^2(\tau_0) |
%\end{eqnarray}  
%
\newline Let us take an example with the periodic function :
%
\begin{equation}\label{funGt}
g(t) = A  \cos(t), \qquad t \in [-\pi,\pi], \qquad A \in \mathbb{R}
\end{equation}
%
We are interested in the development in Taylor-Series of the energy of $g$ following Equation \eqref{EquationEfunction} with $\tau$ and $\tau_0$ in $[-\pi,\pi]$  such that:
%
\begin{eqnarray}
\mathcal{E}(g(\tau))&=& \mathcal{E}(g(\tau_0)) +g^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} \Psi_1^+(g)(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
\end{eqnarray}
% 
%
In addition, with the definition of $\Psi_1^+$ (e.g. Equation \eqref{Psik+defdefef}) one can write:
%
\begin{eqnarray}\label{cossinexample}
\Psi_1^+(g(t)) &=& -2A \cos(t) \sin(t)\nonumber \\
\partial_t \Psi_1^+(g(t)) &=& -2A (\cos^2(t) - \sin^2(t)) \nonumber \\
                          &=& 2A (2\sin^2(t)-1) \nonumber \\
\partial_t^2 \Psi_1^+(g(t)) &=& 8A \sin(t) \cos(t) \nonumber \\ 
\partial_t^3 \Psi_1^+(g(t)) &=& -8A (2\sin^2(t)-1) \nonumber \\ 
\end{eqnarray}
%
One can deduce a general formula for the derivatives of $g$ from the above equations as:
%
%\begin{equation}
\begin{equation}
\forall k \in \mathbb{Z}^+,\left\{
\begin{array}{rcl}
 \partial_t^{2k+1} \Psi_1^+(g(t)) &=& (-1)^{k+1} 2^{2k+1} A (\cos^2(t) - \sin^2(t))\\
 \partial_t^{2k} \Psi_1^+(g(t)) &=& (-1)^{k+1} 2^{2k+1} A (\cos(t) \sin(t))\\
\end{array} \right.
\end{equation}
%\end{equation}
%
From those equations and the general trigonometric properties of the functions $cosines$ and $sines$, the upper bound of $\partial_t^{p} \Psi_1^+(g(t))$ is:
%
\begin{equation}
\forall k \in \mathbb{Z}^+,\left\{
\begin{array}{rcl}
 \partial_t^{2k+1} \Psi_1^+(g(t)) &\leq& | 2^{2k+1} A | \\
 \partial_t^{2k} \Psi_1^+(g(t)) &\leq& | 2^{2k+1} A | \\
\end{array} \right.
\label{example02eq}
\end{equation}
%\begin{equation}
%p \in \mathbb{Z}^+, \qquad |\partial_t^{p} \Psi_1^+(g(t))| \leq |2^{2p+1} A| 
%\end{equation}
%
Let us now examine the convergence of the Taylor-Series of $\mathcal{E}(g)$ using the ratio test (see \cite{Kreyszig}) as described in Equation \eqref{equationenergyseries1}. One can write for $p$ in $\mathbb{Z}^+-\{0,1\}$:
%
\begin{eqnarray}
lim_{p\rightarrow + \infty}\big| \frac{\partial_t^{p+1} (\Psi_1^+(g)(\tau_0) }{\partial_t^{p} \Psi_1^+(g)(\tau_0) } \big| &=& \nonumber \\ 
lim_{p\rightarrow + \infty}\big| 2^2 \frac{(\tau-\tau_0)}{p+1}\big| &=& 0 
\end{eqnarray}
%
Thus, the Taylor-Series of $\mathcal{E}(g)$ is absolutely convergent. Moreover, we can find the distances between the successive derivatives of $\partial_t^{p} \Psi_1^+(g)(\tau_0)$ with $\tau$ and $\tau_0$ in $[-\pi,\pi]$.
%
\begin{eqnarray}
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{p} (\Psi_1^+(g)(\tau_0)| &=& \big( 2^2 \frac{|\tau-\tau_0|}{p+1} -1 \big) |\partial_t^{p} (\Psi_1^+(g)(\tau_0)| \nonumber \\
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{2} (\Psi_1^+(g)(\tau_0)| & =& \sum_{i=2}^{p+1} \big( 2^2 \frac{|\tau-\tau_0|}{i+1} -1 \big) |\partial_t^{i} (\Psi_1^+(g)(\tau_0)|\nonumber \\
%& \leq & \sum_{i=2}^{p+1} \big( 2^2 \frac{|\tau-\tau_0|}{i+1} -1 \big)|2^{i+1} A|\nonumber \\
\end{eqnarray}
%
With Equations \eqref{example02eq}, the distance can be upper bounded as:
%
\begin{equation}
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{2} (\Psi_1^+(g)(\tau_0)| \left\{
\begin{array}{rl}
\leq & \sum_{k=1}^{(p+1)/2} \big( 2^2 \frac{|\tau-\tau_0|}{2 k+1} -1 \big) | 2^{2k+1} A | , \qquad p =2k\\
\leq & \sum_{k=1}^{p/2} \big( 2^2 \frac{|\tau-\tau_0|}{2 k+2} -1 \big) | 2^{2k+1} A | , \qquad p= 2k+1
\end{array} \right.
\end{equation}
%
Note that the distance for the case $p =\{0,1\}$ should be calculated with the equations in \eqref{cossinexample}.
%
%\subsection{Relationship with Parseval-Dirichelet formula} 
%%
%Let us develop Equations \eqref{EquationEfunction} for $t$ in $[\tau, \tau+\epsilon]$ $\subseteq$ $\mathbb{R}$:
%%
%\begin{eqnarray}\label{EquationEfunctionb}
%\mathcal{E}(f_1(\tau + \epsilon)) - \mathcal{E}(f_1(\tau))&=&  f_1^2(\tau) (\epsilon) + \sum_{k=2}^\infty \partial_t^{k-2} \Psi_1^+(f_1)(\tau) \frac{(\epsilon)^k}{k!} \nonumber \\
%\end{eqnarray} 
%%
%If $f_1$ is $T$-periodic, one can write the Parseval-Dirichelet formula such as:
%%
%\begin{eqnarray}
%\mathcal{E}(f_1(\tau)) &=& \frac{1}{T}\int_{\tau-\frac{T}{2}}^{\tau+\frac{T}{2}}f_1^2(t)dt \nonumber \\
%\mathcal{E}(f_1(\tau)) &=& a_0^2(\tau) +\frac{1}{2} \sum_{n=1}^{\infty}(a_n^2(\tau) + b_n^2(\tau))
%\end{eqnarray} 
%%
%With $a_0$, $a_n$ and $b_n$ defined as:
%%
%\begin{eqnarray}
%%\mathcal{E}(g(\tau)) &=& \int_{\tau-\pi}^{\tau+\pi} g^2(t) dt
%a_0 &=& \frac{2}{T}\int_{\tau-\pi}^{\tau+\pi} f_1(t)dt \nonumber \\
%a_n(\tau) &=& \frac{1}{T} \int_{\tau-\pi}^{\tau+\pi} f_1(t)\cos(nt) dt\nonumber \\  
%b_n(\tau) &=& \frac{1}{T} \int_{\tau-\pi}^{\tau+\pi} f_1(t)\sin(nt) dt\nonumber \\
%\end{eqnarray} 
%%
%Thus, one can write:
%%
%\begin{eqnarray}
%\mathcal{E}(f_1(\tau + \epsilon)) - \mathcal{E}(f_1(\tau)) &=& a_0^2(\tau +\epsilon) - a_0^2(\tau) +\frac{1}{2} \sum_{n=1}^{\infty}[(a_n^2(\tau +\epsilon) -a_n^2(\tau)) + (b_n^2(\tau +\epsilon) -b_n^2(\tau))] \nonumber \\
%\end{eqnarray} 
%%
%By identification with the Equation \eqref{EquationEfunctionb}, one can calculate the decomposition of the energy function with the harmonics of $f_1$. Let us then apply this development to the function $g$ defined in the section before with Equation \eqref{funGt} such as:
%%
%\begin{eqnarray}
%%\mathcal{E}(g(\tau)) &=& \int_{\tau-\pi}^{\tau+\pi} g^2(t) dt
%a_0 &=& \frac{1}{2\pi}\int_{\tau-\pi}^{\tau+\pi} A\cos(t)dt \nonumber \\
%a_0 &=& \frac{A}{2\pi}[\sin(t)]_{\tau-\pi}^{\tau+\pi}\nonumber \\
%a_0 &=& 0 \nonumber \\
% & & \nonumber \\
%a_n(\tau) &=& \frac{A}{2\pi} \int_{\tau-\pi}^{\tau+\pi} \cos(t)\cos(nt) dt\nonumber \\  
%b_n(\tau) &=& \frac{A}{2\pi} \int_{\tau-\pi}^{\tau+\pi} \cos(t)\sin(nt) dt\nonumber \\
%\end{eqnarray} 
%%
\section{Conclusions}
%
This work defined two families of DEOs ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$($k \in \mathbb{Z}$).
% defined in time and later on extended in space.
A lemma and theorem were developed to decompose any function $f$ in $\mathbf{S}^{-}(\mathbb{R})$ and its $n$-th power ($n$ in $\mathbb{Z}$ and $n \neq 0$) using the DEO families ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$. In the demonstrations, some close-form formulas are demonstrated such as the decomposition of $f^2$ and $f^3$ with the DEO families ${\Psi}_{k}^{+}$ alone (Lemma) or with  ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ (Theorem) if $f$ is chosen in the subset $\mathbf{s}^{-}(\mathbb{R})$. In addition, the theorem shows the existence and uniqueness of the decomposition with the energy operator families, whereas the lemma only shows the existence of the decomposition when using only  $({\Psi}_{k}^{+})_{k \in \mathbb{Z}}$. Note that the lemma and theorem justify the decomposition with the family of energy operators in the case of $f^n$ with $n \in \mathbb{Z}$ and $n> 1$. However, the discussions following the proof of the theorem deal with the cases $n <0$ and $n=1$.
In the following section, the study of the images and kernels helped to simplify some of the formulas decomposing $f^2$ with ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$. The last part applied this to the energy function and how ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ can appear in the development in Taylor-Series of the energy function.
%
%The corollary of Lemma $2$ shows that the DEOs family can be decomposed with the DEOs family ${\Psi}_{k}^{-}$ for any function $f$ in $Ker(\mathbf{E})$.
\newline This work is an extension of the recently published method to decompose the wave equation with energy operators and a necessary step to extend the method to other linear partial differential equations. 
%
% use section* for acknowledgement
\section*{Acknowledgment}
%
A special thanks is addressed to Professor Alan McIntosh and Dr. Pierre Portal at the Centre for Mathematics and its Applications at the Australian National University for their inputs and discussions when writing this manuscript. The author also acknowledges the comments from Dr. Ryan Loxton from the Department of Mathematics and Statistics at Curtin University, and from Dr. Herb McQueen from the Research School of Earth Sciences at the Australian National University.
% 
% This work is supported by the ARC-SI$2$ Project.
%
% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.
%\ifCLASSOPTIONcaptionsoff
%\newpage
%\fi
% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)

%\bibliographystyle{IEEEtran}
%\bibliography{IEEEabrv,thesis}
\begin{thebibliography}{9}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
%
%\bibitem{Pain2005}H.~J.~Pain, \emph{The Physics of Vibrations and Waves}, 6th~ed.\hskip 1em plus0.5em minus 0.4em\relax John Wiley \& Sons, Ltd, Chichester England, 2005.
%
%\bibitem{Mitra et Al1991}Sanjit~K.~Mitra, Hui~Lui, Ing-Song~Lin and Tian-hu~Yu, \emph{A New Class of Nonlinear Filters for Image Enhancement}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1991 (ICASSP-91), Vol.4, pp. 2525-2528.
% 
%\bibitem{Kaiser et al.1993}R.~B.~Dunn, T.~F.~Quatieri, and J.~F.~Kaiser, \emph{Detection of transient signals using the energy operator}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 145-148.
%
\bibitem{Boudraa et al.2009}
A.~O.~Boudraa, J.~C.~Cexus, S.~Benramdane, T.~Chonavel, \emph{Some useful properties of cross-PsiB-energy operator}, International Journal of Electronics and Communications, vol. 63, pp. 728-735, 2009.
%
\bibitem{Bovik93}
Alan~C.~Bovik, Joseph~P.~Havlicek, and Mita~D.~Desai, \emph{Theorems for Discrete Filtered Modulated Signals}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 153-156.  
%
\bibitem{Hamila1999}
R.~Hamila, M.~Renfors, G.~Gunmarsson, M.~Alanen, \emph{Data processing for mobile phone positioning}, in Proc. IEEE Int. Vehicular Technology Conference, 1999. VTC 1999 - Fall. IEEE VTS 50th, vol.1, pp. 446-449.
%
\bibitem{HamilaJIEEE99}
R.~Hamila, J.~Astola, F.~Alaya~Cheikh, M.~Gabbouj, and M.~Renfors, \emph{Teager Energy and the Ambiguity Function}, IEEE Transactions on Signal Processing, 1999, Vol. 47, No. 1, pp. 260-262.
%
%\bibitem{Kandia2006}
%V.~Kandia, Y.~Stylianou, \emph{Detection of sperm whale clicks based on the Teager-Kaiser energy operator}, International workshop on detection and localization of marine mammals using passive acoustics (No2), Monaco, 2006, vol. 67, pp. 1144-1163.  
%
\bibitem{Kaiser90}
J.~F.~Kaiser, \emph{On a simple algorithm to calculate the 'energy' of a signal}, in Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP-90), vol. 1, pp. 381-384.  
%
\bibitem{Kreyszig} E.~Kreyszig,\emph{Advanced Engineering Mathematics}, Wiley publisher, 8-th Edition, 1998.
%\bibitem{Kaiser93}
%J.~F.~Kaiser, \emph{Some useful properties of Teager's energy operators}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 149-152.  
%
\bibitem{Lin95}
Wei~Lin, C.~Hamilton, P.~Chitrapu, \emph{A generalization to the Teager-Kaiser energy function and application to resolving two closely-spaced tones}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1995 (ICASSP-95), Vol.3, pp. 1637-1640.  
%
\bibitem{Maragos1995}
P.~Maragos and A.~Potamianos, Higher Order Differential Energy Operators, IEEE Signal Processing Letters, vol. 2, No 8, 1995, pp 152-154.
%
\bibitem{JPMontillet2010}
J.P.~Montillet, \emph{On a novel approach to decompose finite energy functions by energy operators and its application to the general wave equation}, International Mathematical forum, 2010, issue 5, no 48, pp. 2387-2400. 
%
\bibitem{BruceWest}
B.J.~West, M.~Bologna, P.~Grigolini, \emph{Physics of Fractal Operators}, 1st Edition, Springer-Verlag New-York, 2003. 

\end{thebibliography}

% biography section
% 
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{biography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:
%
%
%% insert where needed to balance the two columns on the last page with
%% biographies
%\newpage
%%% if you will not have a photo at all:
%%\begin{IEEEbiographynophoto}
%%\begin{biography}{Jean-Philippe Montillet}
%
%\begin{center}
%\begin{large}
%\textbf{Biography}
%\end{large}
%\end{center}
%%
%Jean-Philippe Montillet holds a Master degree in Telecoms
%degree from the Ecole Centrale d� Electronique in Paris
%(France) and a Master degree of Sciences in Electronics
%and Electrical Engineering from Aalborg University (Denmark)
%in 2003. He also holds a Ph.D. from thee University of Nottingham (Uk) [
%the Institute of Engineering Surveying and Space Geodesy
%(IESSG)]. He works
%in the field of precise positioning in urban canyons for civil
%engineering and location based services applications. His
%research interests include statistical signal processing, array
%processing, adaptive filtering, indoor channel models,
%Kalman-filtering, satellite positioning technology and integration of various constellations.
%%
%\end{biography}
%\end{IEEEbiographynophoto}
%
%
% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}

1208.3385.extracted_bib
\begin{thebibliography}{9}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
%
%\bibitem{Pain2005}H.~J.~Pain, \emph{The Physics of Vibrations and Waves}, 6th~ed.\hskip 1em plus0.5em minus 0.4em\relax John Wiley \& Sons, Ltd, Chichester England, 2005.
%
%\bibitem{Mitra et Al1991}Sanjit~K.~Mitra, Hui~Lui, Ing-Song~Lin and Tian-hu~Yu, \emph{A New Class of Nonlinear Filters for Image Enhancement}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1991 (ICASSP-91), Vol.4, pp. 2525-2528.
% 
%\bibitem{Kaiser et al.1993}R.~B.~Dunn, T.~F.~Quatieri, and J.~F.~Kaiser, \emph{Detection of transient signals using the energy operator}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 145-148.
%
\bibitem{Boudraa et al.2009}
A.~O.~Boudraa, J.~C.~Cexus, S.~Benramdane, T.~Chonavel, \emph{Some useful properties of cross-PsiB-energy operator}, International Journal of Electronics and Communications, vol. 63, pp. 728-735, 2009.
%
\bibitem{Bovik93}
Alan~C.~Bovik, Joseph~P.~Havlicek, and Mita~D.~Desai, \emph{Theorems for Discrete Filtered Modulated Signals}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 153-156.  
%
\bibitem{Hamila1999}
R.~Hamila, M.~Renfors, G.~Gunmarsson, M.~Alanen, \emph{Data processing for mobile phone positioning}, in Proc. IEEE Int. Vehicular Technology Conference, 1999. VTC 1999 - Fall. IEEE VTS 50th, vol.1, pp. 446-449.
%
\bibitem{HamilaJIEEE99}
R.~Hamila, J.~Astola, F.~Alaya~Cheikh, M.~Gabbouj, and M.~Renfors, \emph{Teager Energy and the Ambiguity Function}, IEEE Transactions on Signal Processing, 1999, Vol. 47, No. 1, pp. 260-262.
%
%\bibitem{Kandia2006}
%V.~Kandia, Y.~Stylianou, \emph{Detection of sperm whale clicks based on the Teager-Kaiser energy operator}, International workshop on detection and localization of marine mammals using passive acoustics (No2), Monaco, 2006, vol. 67, pp. 1144-1163.  
%
\bibitem{Kaiser90}
J.~F.~Kaiser, \emph{On a simple algorithm to calculate the 'energy' of a signal}, in Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP-90), vol. 1, pp. 381-384.  
%
\bibitem{Kreyszig} E.~Kreyszig,\emph{Advanced Engineering Mathematics}, Wiley publisher, 8-th Edition, 1998.
%\bibitem{Kaiser93}
%J.~F.~Kaiser, \emph{Some useful properties of Teager's energy operators}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1993 (ICASSP-93), Vol.3, pp. 149-152.  
%
\bibitem{Lin95}
Wei~Lin, C.~Hamilton, P.~Chitrapu, \emph{A generalization to the Teager-Kaiser energy function and application to resolving two closely-spaced tones}, in Proc. IEEE Int. Conference on Accoustics, Speech, and Signal Processing, 1995 (ICASSP-95), Vol.3, pp. 1637-1640.  
%
\bibitem{Maragos1995}
P.~Maragos and A.~Potamianos, Higher Order Differential Energy Operators, IEEE Signal Processing Letters, vol. 2, No 8, 1995, pp 152-154.
%
\bibitem{JPMontillet2010}
J.P.~Montillet, \emph{On a novel approach to decompose finite energy functions by energy operators and its application to the general wave equation}, International Mathematical forum, 2010, issue 5, no 48, pp. 2387-2400. 
%
\bibitem{BruceWest}
B.J.~West, M.~Bologna, P.~Grigolini, \emph{Physics of Fractal Operators}, 1st Edition, Springer-Verlag New-York, 2003. 

\end{thebibliography}