12 research outputs found
The Generalization of the Decomposition of Functions by Energy Operators
This work starts with the introduction of a family of differential energy
operators. Energy operators (, ) 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 (,
, k = {0,1,2,..}). The main part of the work is to demonstrate that
for any smooth real-valued function f in the Schwartz space (), the
successive derivatives of the n-th power of f (n in Z and n not equal to 0) can
be decomposed using only (Lemma) or with , (k in
Z) (Theorem) in a unique way (with more restrictive conditions). Some
properties of the Kernel and the Image of the energy operators are given along
with the development. Finally, the paper ends with the application to the
energy function.Comment: The paper was accepted for publication at Acta Applicandae
Mathematicae (15/05/2013) based on v3. v4 is very similar to v3 except that
we modified slightly Definition 1 to make it more readable when showing the
decomposition with the families of energy operator of the derivatives of the
n-th power of
Multi-dimensional higher order differential operators derived from the Teager-Kaiser energy tracking function
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