## The Generalization of the Decomposition of Functions by Energy operators (Part II) and some Applications    [PDF]

J. P. Montillet
This work introduces the families of generalized energy operators $([[.]^p]_k^+)_{k\in\mathbb{Z}}$ and $([[.]^p]_k^-)_{k\in\mathbb{Z}}$ ($p$ in $\mathbb{Z}^+$). One shows that with $\bold{Lemma}$ 1, the successive derivatives of $\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq 0$) can be decomposed with the generalized energy operators $\big ([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ when $f$ is in the subspace $\mathbf{S}_p^-(\mathbb{R})$. With $\bold{Theorem}$ 1 and $f$ in $\mathbf{s}_p^-(\mathbb{R})$, one can decompose uniquely the successive derivatives of $\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq 0$) with the generalized energy operators $\big ([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ and $\big ([[.]^p]_k^-\big)_{k\in\mathbb{Z}}$. $\mathbf{S}_p^-(\mathbb{R})$ and $\mathbf{s}_p^-(\mathbb{R})$ ($p$ in $\mathbb{Z}^+$) are subspaces of the Schwartz space $\mathbf{S}^-(\mathbb{R})$. These results generalize the work of [arxiv/1208.3385]. The second fold of this work is the application of the generalized energy operator families onto the solutions of linear partial differential equations. As an example, the theory is applied to the Helmholtz equation. Note that in this specific case, the use of generalized energy operators in the general solution of this PDE extends the results of [montilletIMF45-48-2010]. Finally, this work ends with some numerical examples. In particular, when defining the Poynting vector and intensity with generalized energy operators applied onto the planar electromagnetic waves, this allows to define a linear relationship with the radiation pressure force.
View original: http://arxiv.org/abs/1308.0874