Generalized Functions and Fundamental Solutions of Differential Equations

Abstract

We denote by D= D(R;ⁿ) = C ^∞₀ (R;ⁿ the space of test functions or the space of basic complex-valued functions ϕ(x) ∈ C^∞(ℝⁿ which are equal to zero for |x| > Rwhere R may depend on ϕ. Convergence in D is defined in the following manner. By definition, a sequence ϕ_k → ϕin D if

• 1) the derivatives ϕ ^(a)_k of any order α = (α₁,…, α_n) converge uniformly on ℝⁿ to ϕ_(α)(x); thus, for all a α $\varphi _{k}^{\alpha }\left( x \right) \rightrightarrows {{\varphi }^{\alpha }}\left( x \right), x \in {{\mathbb{R}}^{n}},$

• 2) for some R, independent of k, ${{\varphi }_{k}}\left( x \right) = 0 for \left| x \right| > R.$