Teager-Kaiser energy operator was defined in  and the family of Teager- Kaiser energy operators in  . Many applications in signal processing were found over the past 25 years such as detecting transient signals  , filtering modulated signals  , image processing  . However,  and  introduced
the conjugate Teager-Kaiser energy operator and associated family .
Subsequently using iterations of the Lie Bracket,  defined the generalized con-
jugate Teager-Kaiser energy operators ( ). To abbreviate
the notation, we sometimes use the generic name energy operator in order to refer to the conjugate Teager-Kaiser energy operators and the generalized con- jugate Teager-Kaiser energy operators. Precision is made in the denomination when it is required. Furthermore, the purpose of the energy operators and generalized energy operators was the decomposition of the successive derivatives of a finite energy function ( in ) in the Schwartz space . The generalized energy operators were introduced when decomposing
the successive derivatives of a finite energy function of the form
( in ) in the Schwartz space. It then follows in  and  the definition of Energy Spaces, which are subspaces of the Schwartz Space associated with energy operators and generalized energy operators. This definition was used to define the concept of multiplicity of solutions in  (Theorem 2 and Corollary 1). The idea is to consider those energy spaces and functions associated with them when solving linear PDEs. More precisely, we look for solutions of a nominated linear PDE within those energy spaces (including the space reduced to ). The concept was further developed using the Taylor series of the energy of a solution for a nominated PDE. The work was based on finding when the successive derivatives, defined through the Taylor series coefficients, are also solutions of this particular PDE (see Section 4 in  ).
This work first generalizes in ( ) the theorems and lemmas established in  and  stated for using the properties of the space called here ( ) together with the general property of the Schwartz space ( )  . However, this work imposes the condition of the stability by Fourier transform for any functions in in order to use the Sobolev space(see Appendix I, Definition I.1). Thus, in this work we consider together with its dual: the tempered distributions . Secondly, the energy spaces ( , ) are also redefined as subspaces of . Furthermore, with the definition of the Sobolev spaces, and in particular the Hilbert spaces , it allows to show the inclusion . Then, we finally redefine in the Theorem 3 established in  and the concept of multiplicity of solutions.
The next section together with Appendix I are reminders about some important definitions and properties for the Sobolev spaces, the Schwartz space and the L2-norm. Section 3 deals with the generalization of the work exposed in  and  in and the redefinition of the energy spaces. Section 4 recalls the concept of multiplicity of solutions defined in  and generalized in with Theorem 4. The last section focuses on some applications of this theory. The first application is the wave equation and the discussion of taking into account more solutions from other energy spaces. We then define another concept called energy parallax (i.e. mathematically in Definition 4, see discussion on the physical interpretation in Appendix II) which is directly related to multiplicity of solutions. In order to illustrate this concept, a second example is the variation of energy density in the skin depth of a conductor material. The idea is to show that the variation of energy density can lead to consider multiple derivatives of evanescent waves resulting from the electromagnetic field. The last section is dedicated to the derivation of the Woodward effect  from the Hoyle-Narlikar theory   using the EM energy density and a discussion takes place about the relationship to the presented theory of energy spaces. It leads to a theoretical definition of an Electromagnetic and Gravitational cou- pling (EMG).
2. Definition of L-2 Norm and Schwartz Space
2.1. Notation and Symbols
In this work, several symbols are used. The set of integer numbers is some- times called only for the positive integer such as or (for a space with dimension m). When the integer 0 is not included, it is explicitly mentioned such as . The set of natural numbers is , with only the positive numbers defined as . is the set of real numbers. Also, the Schwartz space is here called which is the notation used in previous works such as  and  . Several notations describe the relationship between spaces such as intersection ( ), union ( ), inclusion ( , inclusion without the equality , inclusion with equality ). Reader can refer to  or advanced mathematical textbooks for more explanations.
2.2. L-2 Norm and Schwartz Space
With the difference in Appendix I and the generalities with the Sobolev spaces, here the analysis focuses on the L-2 norm (p equal to 2 for the norm). It
allows to state the Plancherel identity :
We are here interested in the functions belonging to the Schwartz space . The Schwartz space consists of smooth functions whose derivatives (including the function) are rapidly decreasing (e.g., the space of all bump functions  ). The Schwartz space is defined as (for in   , for in   ):
where , are multi-indices and
Note that one can define with according to  , but we decide to use following the development in the next sections. It is useful for the remainder of the work to remember some properties of the Schwartz functions in .
Properties 1.  Some Properties of .
・ If , then
・ is a dense subspace of ( ).
・ (Stability with Fourier transform) The Fourier transform is a linear isomorphism .
・ If , then is uniformly continuous on .
The proof of those properties are standard results with Schwartz spaces established in many harmonic analysis books (e.g.,   ).
Remark (1) Note that in    , the author used the general term of finite energy functions for Schwartz functions in , with restricted to . It is a common definition in signal processing for the functions in and generally associated with the Plancherel identity.
Remark (2) One way to interpret the property that is stable by Fourier transform is:
Now, let us recall the definition of the Hilbert spaces (Sobolev spaces for , see Appendix I, Definition I.1) from (35) and drop the sup-script p in the remainder of this work:
Note that is the space of tempered distributions, dual of via the Fourier transform. A function belongs to if and only if its Fourier transform belongs to and the Fourier transform preserves the L2-norm. As a result, the Fourier transform provides a simple way to define L2- Sobolev spaces on (including ones of fractional and negative order  ). Finally, the stability via Fourier transform is the key for .
Remark (3) Following the remark (Remark 3.4 in  ) and the general properties of the Fourier transform, one can state the equivalence relationship in
Using the definition of and the properties of the Fourier transform, it is also possible to show that for ,  , and the relationship . It is also possible to define
with , and to extend this equality to following  .
3. On Some Subsets of Schwartz Spaces: Energy Spaces
This section first recalls generalities on the Teager-Kaiser energy operator and its conjugate operator with the application to decompose Schwartz functions from the work developed in  and  . We call in this work Energy operators the families of operators based on the Teager-Kaiser energy operator. The definitions and theorems are here stated for the Schwartz space ( ) whereas the preliminary work in  and  stated the definitions and main theorems for . For in Section 6 in  , a discussion takes place during the application of the theory to linear partial differential equations. Secondly, the energy spaces defined in  and  are here generalized on with novel relationships with Sobolev spaces ( ).
3.1. Definition and Properties of the Energy Operators in
Let us call the set all Schwartz functions (or operators) defined such as . For , let us define
( , ), with defined with the vector parameter such as
Combining multiple integrals and derivatives justify the use of the Schwartz space and echoes the choice made previously in  (see equation (10)). The definitions and results given in  and  in the case are now formulated for . Section 2 in  and Section 4 in  defined the energy operators , ( in ) and the generalized energy operators
and ( in ). Following  , let us define the energy
operators with multi-index derivative in (7):
Further more, we also use the short notation in the remainder of this work. Note that is the conjugate operator of and respec- tively to .
Remark (4) The families of (generalized) energy operators and ( in ) are also called families of differential energy operator
(DEO)   .
Furthermore,  defined the generalized energy operators and ( ):
By iterating the bracket ,  defined the generalized operator and the conjugate with p in . Note that in and i in .
Now, the derivative chain rule property and bilinearity of the energy operators and generalized operators (for i in ) are shown respectively in  , Section 2 and  , Proposition 3. The generalisation of this property to i in for
the operators , , and ( , ) is
trivial due to the linearity of the derivatives and integrals when defining in (7). Due to the linearity of the sum, the bilinearity property is also generalized to
, , and ( , ).
Definition 1.  in , , and , the family of operators (with ) decom- poses in ( ), if it exists , , and it exists and r in (with ) such as .
In addition, one has to define as:
or with the energy operators and defined in (8)
is the notation for the kernel associated here with the operators , , and ( in ) (see  , Properties 1 and 2). By definition, one can state that . Following Definition 1, the uniqueness of the decomposition in with the families of differential operators can be stated as:
Definition 2.  in , , and , the
families of operators and ( and decompose uniquely in , if for any
family of operators decomposing in , there exists a unique couple in such as:
Two important results shown in  are:
Lemma 1 For in , the family of DEO ( ) decomposes , , and .
Theorem 1. For in , the families of DEO and ( ) decompose uniquely , , and .
The Lemma 1 and Theorem 1 were then extended in  to the family of generalized operator with :
Lemma 2. For in , in , the families of generalized energy operators ( ) decompose , and .
Theorem 2. For in , for in , the families of generalized operators and ( ) decompose uniquely , and .
and ( in ) are energy spaces in defined in the next section.
Remark (5) One can extend the Theorem 1, Theorem 2, Lemma 1 and Lemma 2 for with in following previous discussions in  (Section 3, p.74) and  (Section 4). is here restricted to in order to easy the whole mathematical development.
3.2. Energy Spaces in
Let us introduce the energy spaces and some properties.
Definition 3. (  , Definition 3) The energy space , with in , is equal to .
With for in defined as
The energy spaces, and ( ) , cited in Lemma 2 and Theorem 2 are defined:
Remark (6) Definition 3 does not follow completely Definition 3 in  , because the energy space is defined here , and only for in  .
Remark (7) In the previous definition, ( ). Also, , whereas in  and  . The inclusion does not change Lemma 2 and Theorem 2 (i.e. ). The justification of not
including this space was only based on the applications of the theory in  and  which is not justified in this work.
We can now state some properties associated with the energy spaces on .
Properties 2. in , and in particular , in (with ), in , we have the following inclusions:
1) Let us recall the definition of the Hilbert space on according to Appendix I, Definition I.1 and Definition 1.
Looking at the definition of the energy space and , one can notice the similitude. However , the multi-index derivative (  , chap. 1.1)
contains also the cross-derivatives (e.g., ), whereas there are no cross-
derivatives in the definition of at the beginning of Appendix I. Thus, the energy spaces ( , ) is defined without the cross- derivatives. In addition with Properties 1, . Thus, by definition we have the relationship :
2) With Remark (3), we know that for , . Now, with 1), and . Now by definition of and , . Finally, .
3) From Remark (3), , and (by defi-
nition of the energy space) ( ). Thus, ( ).
Furthermore, Appendix III discusses the relationship between the subspaces and ( ). Finally, because we are studying functions and operators in subspaces of with , one need to extend Proposition 1 in  and  .
Proposition 1. If for , and analytic; for any and ( ), and is analytic, where
is a convergent series.
Proof. The proof of Proposition 1 for i equal 1 is given in  (p.4). The extension of the proof for the case i equal m is straightforward with the general definition for any and ( ).
4. Multiplicity of the Solutions in
To recall  , a possible application of the theory of the energy operators is to look at solutions of a given partial differential equation for solutions in of the form . Instead of solving the equation for specific values (e.g., boundary conditions), the work in  (  , Theorem 1 and corollary)defines the concept of multiplicity of solutions in ( ) such as the study of the multiple solutions of a PDE based on the definition of the energy spaces ( ). One way to understand this concept, is to study the convergence of the development in Taylor series of the energy function associated to a nominated energy space. It was shown in  that taking into account additional terms of the Taylor series leads to define additional solutions of the wave equation (see Section 4  ). In this section, we extend this concept to ( ) and we reformulate the results from  for the solutions in the subspaces ( , ).
Let us define any PDEs of the form:
Thus, all the solutions are here defined in . Now, we are interested in the solutions which can be defined on the energy spaces
( ). In other words, . In particular, we choose the solution . Furthermore, one can define , such as for . In other words, and , such as . Now, one can then state a general theorem of multiplicty of solutions based on  . It follows:
Theorem 3. (Multiplicity of Solutions in ) If is a subspace of all the solutions of a nominated linear PDE. For , is in . Then, is a solution for this linear PDE if and only if:
1) (General condition to be a solution) .
2) (Solutions in ) , such as .
3) (Multiplicity of the solutions) If ( ), and
, such as ( ) and , , .
4) (Superposition of solutions and energy conservation ) If , with such as ( ), then .
Proof. The proof is the generalization of what was already written in  (see Theorem 2 in  ) for the case m equal 1. Here is the generalization to m.
1) This is the definition of a solution for a nominated PDE with solutions in and in the energy space .
2) , thus . With Proposition 1, it
means that for any and ( )
Thus, following  , one can define such as and then we define . With our notation, it is equivalent to write .
3) It is sufficient to show that for , , .
Now, with the definition , and . In addition, , ( ) and . The interest
of this statement is the function such as with
and . In particular, if we introduce a numerical approximation in order to get the condition . In other words,
In some examples in Section 4 in  and Section 6 in  , it is shown that the evanescent waves when solving the wave equation for specific solutions, is a particular example of those functions.
4) The proof follows  (Theorem 2). This statement is to guarantee that there is a finite sum of energy with the superposition of multiple solutions. Thus with the development in statement (2.), one can use the Minkowski inequality (e.g,  , Theorem 202) for in ( )
with . Thus, (4.) stands if and only if . As , , is in , it then exists . One possibility is in such as , then
5. Some Applications
This section focuses on the application of the energy space theory. The first section is the study of the concept of multiplicity of solutions with a simple mathematical example using the wave equation. Then, the second section is discussing the application of this concept within the Woodward effect  
5.1. Energy Variation and Wave Equation
As a simple case of linear PDE, the wave equation with the particular solutions of the form of evanescent waves, was already discussed in Section 6 of  and  . However, it is an interesting example to apply and understand the concept of multiplicity stated in Theorem 3. From  , the wave equation can be formulated in (with t and r the time and space variables):
c is the speed of light. Note that the values of t and r are restricted to some interval, because it is conventional to solve the equation for a restricted time interval in and a specific region in space. According to the previous section, we are here interested in the solutions in the energy (sub)space , of the kind
( in , in , in ). Furthermore, the relationship imposes that the solu-
tions should be finite energy functions, decaying for large values of r and t. It was previously underlined in  and  that planar waves should be rejected, because this type of solutions does not belong to . However, evanescent waves are a type of solutions included in and considered in this work. They are here defined such as:
, and are the wave numbers, is the angular frequency and is the amplitude of this wave  . Assuming and ( , ) known, one can add some boundary conditions in order to estimate , and . Furthermore, a traveling wave solution of (19) should satisfy the dispersion relationship between , and  . However, our interest is just the general form assuming that all the parameters are known. For , the type of solutions in are:
In , one can then write the type of solutions
Let us consider the form of solutions which propagates in a closed cavity (e.g., closed wave guide  ). One possible solution is the evanescent wave described in (20). Now, if and are analytic in , with Proprsition 1 we can assume that is finite energy (and more generally in ) with a wise choice on the parameters , , and . One can estimate the difference of energy in time over inside the cavity at a specific location ( in ) such as
Here the symbol ‘ ‘ means that
Now, let us do a hypothesis that increases significantly over modifying the approximation in (24)
To recall that , and , and using Theorem 3, one can take into account solutions in those subspaces. The multiplicity of the solutions due to the variation of energy can be formulated as an approximation for taking into account additional solutions produced by the wave equation.
Remark (8): In  , the general idea was to look for the solutions of linear PDEs in associated with energy subspaces ( ) in order to apply Theorem 1 in  , which is here generalized in Theorem 3 for ( ). The purpose was to find the subspaces reduced to when studying the convergence of the Taylor series of the energy functions. However, the redefinition of the energy subspaces within the Sobolev spaces defined in Section 3 allows us to look for solutions in in order to use Lemma 2. Because of the inclusion of the energy spaces shown in Properties 2 using the Sobolev embedding (e.g., Theorem I.1 in Appendix I) such as ( , ), .
Definition 4. (Energy Parallax) Considering a linear PDE with some solutions in such as . Furthermore, if it exists and such as , then we associate the energy for , such as one can estimate the variation over an elementary quantity (e.g., space or time). If is not negligible ( such as and ), then one can consider additional solutions in .
5.2. Variation of EM Energy Density and the Woodward Effect
In this section, the theory of energy space is applied to the possible variations of electromagnetic energy density due to, for example, skin depth effect  inside some conductive material. Beyond this application, the interest is to give a physical meaning of taking into account those additional solutions in various energy spaces. The second part is dedicated to the Woodward effect and the possible relationship with the variation of EM energy density in some specific settings.
5.2.1. Variation of EM Energy Density
Thus, let us formulate the variation in time of energy density (u) at the second order with a Taylor series development such as:
is the Landau notation to omit higher order quantities. Note that at the
first order . The higher orders term are based on the assumptions that
the EM waves inside the skin layer of the copper plate are evanescent waves and thus functions in the Schwartz space ( -with 3 dimension variables and considering also the time )  . As discussed before, those solutions are finite energy functions and in (i.e. following  and  , at some given point in the skin layer defined by the coordinates ). Now, using the Lemma 1 and the space in Section 3, we can state in
Here is either the electric or magnetic field (i.e. the absolute norm of and respectively). With the concept of multiplicity of solutions (e.g., Theorem 3). If is a general solution of some linear PDEs, then can be identified as a special form of the solution (conditionally to its existence ).
Now considering the wave equation, the electric field and magnetic field are solutions and belong to the subspace and associated with the variation of energy density . Furthermore, we can consider the solutions in associated with the variation of energy density , which can be explained with the concept of multiplicity of solutions. The solutions of interest in are for the electric field and the magnetic field . The Taylor Series development of the energy of (for example) the electric field on a nominated position in space (i.e., ) and in an increment of time :
Finally one can write the relationship with the energy density following (26) and the previous Taylor series development for the electric and magnetic field:
Therefore, taking into account the second order term of the energy density means that additional solutions should also be considered in the EM modeling. Note that in Appendix IV, we are taking an example of evanescent waves inside a copper wall (i.e. skin depth effect  ) and try to give further meaning to the consideration of higher order derivatives of the EM energy density where the additional solutions are defined with the energy spaces (e.g., and in ).
5.2.2. Derivation of the Woodward Effect Using the Electromagnetic Energy Density
This section focuses on the derivation of the Woodward effect created in a asymmetric EM cavity (i.e. frustum) due to EM waves reflected on the cavity’s wall. Thus, the assumption is that the EM energy density variation results from the evanescent waves taking place in the skin depth of the asymmetric EM cavity’s walls.
1) Assumptions with the energy momentum relationship
When the Woodward effect was established in   , the authors implicitly assumed the rest mass of the piezoelectric material via the famous Einstein’s relation in special relativity ( the rest energy associated with the rest mass ) and its variation via electrostrictive effect.
Here, the system is the asymmetric EM cavity. The rest mass is all the particles within it at time when no charges are on the cavity’s walls. It excludes the photons considered with a null mass. Thus, the main assumption is that the EM excitation on the walls creates electric charges (i.e. electrons) which makes the rest mass varying with time. This assumption is the same as the mass variation of a capacitor between the charge and discharge times  . It allows us to state the variation of rest energy such as:
Finally, the variation of rest energy is assumed to be equal to the variation of EM energy density ( ) resulting from the charges within the skin depth of the walls. We neglect any electrostrictive effects compared to the variation of EM energy density.
Note that at the particle level, the rest mass should satisfy the energy momen- tum relationship ( ) for a free body in special relativity  :
with the momentum and the rest mass of the particle associated with the total energy . The particle is accelerated via the Lorentz force applied to the whole cavity with obviously . Thus, we have also the relationship . In the remainder, we also use the elementary variation which becomes for an infinitesimally small variation.
2) Derivation of the Woodward effect and relationship with EM energy density
If we define the mass density such as , then from  , one can write the elementary mass variation per unit of volume
Let us define the the rest energy , then
In some particular cases such as an EM cavity, we assume that the variation in time of the rest energy is equal to the variation of EM energy density (i.e. ), but the rest energy is much bigger than the EM energy density . It allows then to state the relationship between the Woodward effect and the EM energy density
The EM energy density follows the general definition of the sum of energy density from the electric ( ) and magnetic ( ) fields  . Finally, (38) can be seen as the definition of the EMG coupling.
This work generalizes in the Schwartz space , the framework on conjugate Teager-Kaiser energy operators established in  and  for the case m in . The concept of multiplicity of solutions defined in  is also redefined here in Theorem 3. However, this concept uses the notion of energy spaces ( ( , ), subspaces of defined previously in  and  . In order to generalize their definition as subspaces of , the theory has been extended to some properties on the Hilbert spaces ( ) on . In particular, we show in Properties 2 that ( ) and the inclusion .
The concept of multiplicity of solutions focuses on, generally speaking, looking for solutions of a given linear PDE specifically in the energy spaces. In this way, it is not following the classical way of solving a linear PDE with boundary conditions. Three examples illustrate this concept. The first one investigates some type of solutions (e.g., evanescent waves) of the wave equation when analysing the Taylor series development of the energy function associated with an evanescent wave. We then formulate another concept: the energy parallax. It is defined mathematically in Definition 4. Under some specific circumstances (e.g., the energy function exists), we show that the variations of energy locally in a predefined system, should lead to include additional solutions in the energy spaces with higher order v (in ). The second example is based on the local variations of EM energy density, which allows to define waves which are first order derivative of the EM field. This example is further explored in Appendix IV. Finally, the last example is the derivation of the Woodward effect with some strong hypothesis in order to include the EM energy density in the specific case of asymmetric EM cavity. We introduce in the Woodward effect, the first and second order derivative of the EM energy density, which can be interpreted such as a theoretical definition of an Electromagnetic and Gravita- tional coupling.
The author would like to acknowledge the important discussions with Dr. José Rodal and Prof. Heidi Fearn (California State University Fullerton, physics department) on the Woodward effect and its derivation from general relativity. In addition, we thank Prof. Paul Jolissaint (Université de Neuchâtel, Institute of Mathematics) for his kind advices on the Sobolev spaces, and Prof. Marc Troyanov together with Dr. Luigi Provenzano (Ecole Polytechnique Fédérale de Lausanne, Institute of Mathematics) for the corrections on the Sobolev spaces and the Schwartz space. Finally, we acknowledge the constructive comments from the anonymous reviewers improving this work.
Appendix I: Generalities on Sobolev Spaces
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev.
Definition I.1.  Let ( ) be open. The Sobolev space ( , ) is defined as:
with the a-th partial derivative in multi index notation,
. The Sobolev space is the space of all locally
integrable functions in such as their partial derivatives exist in the weak sense for all multi index and belongs to (i.e. ) (  , chap. 5). If lies in , we define the norm of by the formula
Now, let us introduce the Fourier transform as in 
Here is the space of bounded and continuous functions in  . Note that is the scalar product (with and in ). One can then define the Sobolev spaces for , using the Bessel potentials and the Fourier transform such as  or  (chap. 9) :
The Bessel potential spaces are defined when replacing by any real number . They are Banach spaces and, for the special case , Hilbert spaces . Now, one can state an important result with Sobolev spaces 
Theorem I.1.: , whenever and are such that
Proof. The proof of this theorem is rather long and technically delicate which is not our focus. Readers interested in this matter should refer to   (chap. 5)
Appendix II: Possible Interpretation of the Energy Parallax in Modern Physics
In Section 4, we define mathematically the notion of multiplicity of solutions for a given PDE. Through the various examples in Section 5, we define the concept of energy parallax. The general meaning is that additional solutions should be taken into account when varying the amount of energy. Those solutions should be defined based on the associated energy spaces (e.g., , ). Now, if we replace this concept in modern physics, what is the meaning behind it?
In modern physics, Energy is a global concept across the whole science. The definition varies with for example kinetic energy and potential energy in classical mechanics. It relates respectively to the object’s movement through space and function of its position within a field  . Chemical energy can be defined broadly such as the electrical potential energy among atoms and molecules. In quantum mechanics, energy is defined in terms of energy operators (e.g., Hamil- tonian) as a time derivative of the work function. It allows to define particles at nominated energy levels associated with an EM waves emitted at frequencies defined by the Planck’s relation. In General Relativity, energy results from the product of a varying mass and the square of the speed of light. Energy can describe the behavior of a system of two particles (and more). For example, the electron-positron annihilation in which rest mass (invariant mass) is destroyed. At the opposite, the inverse process (creator) in which the rest mass of the particle is created from energy of two (or more) annihilating photons  .
Energy parallax is here defined such as the concept of using additional wave functions. For example in Section 5.2.2 increasing the higher order derivatives of the EM energy density leads to the consideration of additional waves. The energy parallax concept can then help us to state that those additional waves are additional excited photons that we must take into account to vary the EM energy density.
Appendix III: Discussion on the Possible Relationship between the Energy Spaces and
This section follows the development in Section 3.2 and especially Properties 2. First, , , because , and ( , ) . Thus, .
To recall Definition 2 and Lemma 2, can be decomposed with the family of energy operators ( , , , ). Thus, one can write ( ):
Thus, for , Lemma 2 allows to state that , with the subspace of , but restricted for and .
Furthermore, let us define the space :
Note that , but the bump functions  are included in . We can also recall the discussion on in  and  , with the definition
On can also state that ( ) and use the
Leibniz’s rule for derivations in order to expand the multiple derivatives or the decomposition stated in Lemma 2. If we call ( ), the subspaces of . For all in can be written as a non linear sum of in . Finally, we can conclude that . With the specific extension of to the case , we can also conclude . In addition,
Appendix IV: Consequences in Terms of EM theory
We are taking the example of the variation of EM energy density inside a copper wall due to planar waves reflecting and refracting on it  . To recall Section 5.2, the EM field is now including ( , ) and ( , ), contribution of the subspaces and respectively when using the concept of multiplicity of the solutions (i.e. Theprem 3) for the higher order derivatives of the energy density (see (26)). We call the total EM field and inside the copper plate (skin layer) with associated permittivity and permeability . They are solutions of the Maxwell equations:
with the principle of charge conservation:
Now, the variation of energy density (26) together with the equation of charge conservation is formulated such as:
is the Poynting vector. Now, writing , and is the first derivative in time ( ) (i.e. solutions in ),
then following 
using the equalities and the Maxwell equation , the previous equation reduces to:
We can separate in three groups,
The Poynting vector is defined as and its derivative . Thus, the second order term of the energy density is
the contribution of the EM field generated by and is:
The last line is the contribution from only the fields and .
Finally, the creation of the wave defined by the EM field ( , ) means that some material properties may allow to create two type of EM waves namely ( , ) and ( , ).