For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$ Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$. a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$. b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations. c) Sketch the graphs of $I_2$ and $I_3$. d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points. e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?
For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set
$$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$.\\
a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$.\\
b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations.\\
c) Sketch the graphs of $I_2$ and $I_3$.\\
d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points.\\
e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?