The generalization of the decomposition of functions by energy operators (Q2439467)

Differential energy operators, decomposition of a function by a family of such operators and the uniqueness of this decomposition are to be understood in the sense defined in previous papers, for example, in the author's paper [Int. Math. Forum 5, No. 45--48, 2387--2400 (2010; Zbl 1225.35139)]. The goal of the present paper is to prove a decomposition result. Namely, the author shows that the successive derivatives of the \(n\)-th power of a function \(f\) belonging to a Schwartz space \(S^-(\mathbb{R})\) can be uniquely decomposed by a family of differential energy operators. There follows a discussion on different values of \(n\) and on some properties of the images and kernels of the considered differential energy operators. The result is applied to an energy function.