Q16
Sequences and series, recurrence and convergence
Sequence of functions convergence
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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 $k \in \mathbb{N}$ and $n \geqslant k+2$, we have
$$\left\|\psi_{n+1}^{(k)} - \psi_{n}^{(k)}\right\|_{\infty} \leqslant \frac{\mu_{n+1}}{2} \left\|\psi_{k+1}^{(k+1)}\right\|_{\infty}.$$
Deduce that for all $k \in \mathbb{N}$, the sequence of functions $\psi_{n}^{(k)}$ converges uniformly on $\mathbb{R}$.