Define the Schwartz space $\mathcal{S}(\mathbb{R})$ and the corresponding space of tempered distributions $\mathcal{S}^{\prime}(\mathbb{R})$.

Use the Fourier transform to give an integral formula for the solution of the equation

$$-\frac{d^{2} u}{d x^{2}}+\frac{d u}{d x}+u=f$$

for $f \in \mathcal{S}(\mathbb{R})$. Prove that your solution lies in $\mathcal{S}(\mathbb{R})$. Is your formula the unique solution to $(*)$ in the Schwartz space?

Deduce from this formula an integral expression for the fundamental solution of the operator $P=-\frac{d^{2}}{d x^{2}}+\frac{d}{d x}+1$.

Let $K$ be the function:

$$K(x)= \begin{cases}\frac{1}{\sqrt{5}} e^{-(\sqrt{5}-1) x / 2} & \text { for } x \geqslant 0 \\ \frac{1}{\sqrt{5}} e^{(\sqrt{5}+1) x / 2} & \text { for } x \leqslant 0\end{cases}$$

Using the definition of distributional derivatives verify that this function is a fundamental solution for $P$.