For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Show that $\delta(f)$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}_+^*$. Compare $(\delta(f))'$ and $\delta(f')$.

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \geqslant 1$ we have
$$\left\|\psi_{k}^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{0}\right\|_{\infty} \frac{1}{\mu_{1} \cdots \mu_{k}}$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Let $f = \lim_{n \rightarrow \infty} \psi_{n}$. Show that $f$ has compact support and that if $\psi_{0}$ is positive and not identically zero, then so is $f$.

We seek to show that if $(M_{n})_{n \geqslant 0}$ is a sequence of strictly positive real numbers such that the series $\sum_{n \geqslant 1} \frac{M_{n-1}}{M_{n}}$ converges, there exists a function $f \in \mathcal{C}_{c}^{\infty}(\mathbb{R})$ not identically zero such that for all $n \geqslant 0$, $\|f^{(n)}\|_{\infty} \leqslant M_{n}$.
Conclude regarding the initially posed question.

Show that for all functions $f, g \in C^{2}(\mathbf{R})$ such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x) g(x) \varphi(x) \mathrm{d}x = -\int_{-\infty}^{+\infty} f^{\prime}(x) g^{\prime}(x) \varphi(x) \mathrm{d}x$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$

Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have
$$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$

Show that for all functions $f, g \in C^2(\mathbf{R})$ such that the functions $f, f', f''$ and $g$ have slow growth, we have $$\int_{-\infty}^{+\infty} L(f)(x)\,g(x)\,\varphi(x)\,\mathrm{d}x = -\int_{-\infty}^{+\infty} f'(x)\,g'(x)\,\varphi(x)\,\mathrm{d}x,$$ where $L(f)(x) = f''(x) - x f'(x)$ and $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$.

Show that if $f \in C^1(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f' \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbf{R}_+ \mapsto P_t(f)(x)$ is of class $C^1$ on $\mathbf{R}_+^*$ and show that for all $t > 0$, we have $$\frac{\partial P_t(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x\mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1-\mathrm{e}^{-2t}}}\,y\right) f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$