Fundamental Matrices of Homogeneous Systems

In this last chapter we exploit the homogeneity of a system in order to reduce the number of integrations in the representation of its fundamental matrix by inverse Fourier transform. Let us roughly sketch the idea. If P(∂) is an elliptic and homogeneous operator of degree m, we obtain for a fundamental solution E the following: $$\displaystyle{E(x) = \mathcal{F}^{-1}{\Bigl ( \frac{1} {P(\text{i}\xi )}\Bigr )} = \text{i}^{-m}(2\pi )^{-n}\langle 1_{\xi },\text{e}^{\text{i}x\xi }P(\xi )^{-1}\rangle.}$$ Upon introducing polar coordinates $$\xi = r\omega$$ this yields $$\displaystyle\begin{array}{rcl} E(x)& =& \text{i}^{-m}(2\pi )^{-n}\langle r_{ +}^{n-1}\vert \gamma \vert (\omega ),\text{e}^{\text{i}x\omega r}P(r\omega )^{-1}\rangle {}\\ & =& \text{i}^{-m}(2\pi )^{-n}\langle \langle \vert \gamma \vert (\omega ),P(\omega )^{-1}\langle r_{ +}^{n-m-1},\text{e}^{\text{i}x\omega r}\rangle \rangle {}\\ & =& \text{i}^{-m}(2\pi )^{-n}\langle \langle \vert \gamma \vert (\omega ),P(\omega )^{-1}\mathcal{F}(t_{ +}^{n-m-1})(-x\omega )\rangle {}\\ \end{array}$$ Since $$\mathcal{F}(t_{+}^{n-m-1})$$ is explicitly known from elementary distribution theory, this yields a very symmetrical formula for E, see Proposition 5.2.1 It is, however, of limited practical value for the calculation of E, see Example 5.2.2.