Fourier Transformation

Abstract

We show that the Fourier transformation is an isomorphism of the topological vector space of tempered distributions ${\mathcal{S}}^{\prime}(\mathbb{R}^n)$. This makes the Fourier transformation a powerful tool of analysis, in particular in the solution theory for constant coefficient partial differential operators P(D). The proof of this result is prepared by establishing that the Fourier transformation is an isomorphism of the topological vector space ${\mathcal{S}}(\mathbb{R}^n)$. We determine the image of various operations on test functions (differentiation, translation, multiplication with the independent variable) under the Fourier transformation. By duality these relations carry over to the Fourier transformation on tempered distributions. According to the convolution theorem the Fourier transformation maps a convolution product into a pointwise product and vice versa. Application of the Fourier transformation to the constant coefficient partial differential equation $P(D)T=S$ for given $S \in{\mathcal{S}}^{\prime}(\mathbb{R}^n)$ leads to the algebraic equation for the Fourier transforms $P(\;\textrm{i}\xi)\tilde{T}= \tilde{S}$. Hörmander’s theorem ensures that this algebraic equation always has a tempered solution and thus implies that the given partial differential equation can be solved. As a further application of the Fourier transformation we determine again the elementary solution of some well known partial differential operators.