Gâteaux Differentiability of the Polynomial Test and Generalized Functions

Abstract

Let S₊ and S₊^′ be the Schwartz spaces of rapidly decreasing functions and tempered distributions on ℝ₊ , respectively. Let P(S₊^′) be the space of continuous polynomials over S₊^′ and let P ′ (S₊^′) be its strong dual. These spaces have representations in the form of Fock type spaces

\( \varGamma \left({S}_{+}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}\right)\kern2em \mathrm{and}\kern2em \varGamma \left({S}_{+}^{\prime}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\times}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}^{\prime}\right), \)

respectively. In the present paper, the Gâteaux differentiability of elements of the spaces P(S₊^′), P ′ (S₊^′), Γ(S₊), and Γ(S₊^′) is investigated. The relationship between the Gâteaux derivative, the operators of creation and annihilation in the Fock type spaces, and the differentiations on Γ(S₊) and Γ(S₊^′) is established.