For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by $$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$ and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula $$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$ a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity. b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$. c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.

Does there exist a function $g \in C_{b}(\mathbb{R})$ such that for every function $f$ in $L^{1}(\mathbb{R})$, we have $f * g = f$?

For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by $$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$ For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.

Let $f, g \in L^{1}(\mathbb{R})$. Assume that $g$ is bounded. a) Show that $f * g$ is integrable on $\mathbb{R}$ and determine $\int_{\mathbb{R}} f * g$ in terms of $\int_{\mathbb{R}} f$ and $\int_{\mathbb{R}} g$. b) Show that $\widehat{f * g} = \hat{f} \times \hat{g}$.

Show that there exist two functions $f$ and $g$ in $L^{1}(\mathbb{R})$ such that $f * g(0)$ is not defined.

We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ Express the Fourier transform $\hat{k}_{n}(x)$ using the function defined by $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$

We define $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Justify that $\varphi \in L^{1}(\mathbb{R})$.

We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ We admit that $\int_{\mathbb{R}} \varphi = \pi$ where $\varphi(x) = \left(\frac{\sin x}{x}\right)^2$ for $x\neq 0$ and $\varphi(0)=1$. We set $K_{n} = \frac{1}{2\pi} \hat{k}_{n}$. Show that the sequence of functions $\left(K_{n}\right)_{n \geqslant 1}$ is an approximate identity.

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ For every real $t$ and every non-zero natural number $n$, show that $I_{n}(t) = \left(f * K_{n}\right)(t)$.

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ Deduce, for every real $t$: $$f(t) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(x) \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}x$$

To any function $g$ in $C(\mathbb{R})$, we associate the linear form $\varphi_{g}$ on $L^{1}(\mathbb{R})$ defined by $$\varphi_{g}(f) = \int_{\mathbb{R}} f(t) g(-t) \mathrm{d}t$$ Let $(g_{1}, \ldots, g_{p})$ be a family of elements of $C_{b}(\mathbb{R})$. Show that the family $(g_{1}, \ldots, g_{p})$ is free if and only if the family $(\varphi_{g_{1}}, \ldots, \varphi_{g_{p}})$ is free.

Let $E$ be an infinite-dimensional vector space and $\left(f_{n}\right)_{n \in \mathbb{N}}$ a family of linear forms on $E$. We denote $$K = \bigcap_{n \in \mathbb{N}} \operatorname{Ker}\left(f_{n}\right)$$ Show that the codimension of $K$ in $E$ is equal to the rank of the family $\left(f_{n}\right)_{n \in \mathbb{N}}$ in the dual space $E^{*}$ (begin with the case where this rank is finite).

We assume that $g \in C_{b}(\mathbb{R})$. We consider the vector subspace $$N_{g} = \left\{f \in L^{1}(\mathbb{R}) \mid f * g = 0\right\}$$ and the vector space $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ where $T_{\alpha}(g)(x) = g(x-\alpha)$. Show that the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$ is equal to the dimension of $V_{g}$.

a) Let $\beta \in \mathbb{R}$ and let $g$ be the function defined by $g(t) = \mathrm{e}^{\mathrm{i}\beta t}$ for all $t \in \mathbb{R}$. Determine the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$. b) Let $n$ be a natural number. Show that there exists a function $g$ in $C_{b}(\mathbb{R})$ such that $N_{g}$ has codimension $n$ in $L^{1}(\mathbb{R})$.

Let $g \in C_{b}(\mathbb{R})$. We assume that $N_{g}$ has finite codimension $n$ in $L^{1}(\mathbb{R})$, and that $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ has dimension $n$. Show that there exist real numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ and functions $m_{1}, \ldots, m_{n}$ of a real variable such that, for every real $\alpha$, $$T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$$

Let $F$ be a finite-dimensional subspace of $C(\mathbb{R})$, with dimension denoted $p$. For any function $f \in C(\mathbb{R})$ and for any real $x$, we denote $e_{x}(f) = f(x)$. a) Show that there exist real numbers $a_{1}, \ldots, a_{p}$ such that $(e_{a_{1}}, \ldots, e_{a_{p}})$ is a basis of the dual space $F^{*}$. b) If $\left(f_{1}, \ldots, f_{p}\right)$ is a family of elements of $F$, show that $\operatorname{Det}\left(f_{i}\left(a_{j}\right)\right)_{1 \leqslant i,j \leqslant p}$ is non-zero if and only if $\left(f_{1}, \ldots, f_{p}\right)$ is a basis of $F$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $\alpha_{1}, \ldots, \alpha_{n}$, $m_{1}, \ldots, m_{n}$ be as in III.C.1 such that $T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$ for every real $\alpha$. By applying question III.C.2) to $V_{g}$, show that if $g$ is of class $C^{k}$ then the functions $m_{1}, \ldots, m_{n}$ are of class $C^{k}$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3, defined on $[-1,1]$ by $h_{r}(t) = \frac{(1-t^2)^r}{\lambda_r}$ and zero outside $[-1,1]$. Show that for every non-zero natural number $r$, $V_{h_{r} * g}$ is finite-dimensional.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3. Show that for $r$ sufficiently large the dimension of $V_{h_{r} * g}$ is equal to that of $V_{g}$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $m_{1}, \ldots, m_{n}$ be as in III.C.1. Deduce that the functions $m_{1}, \ldots, m_{n}$ are of class $C^{\infty}$.

Determine the set of functions $g \in C_{b}(\mathbb{R})$ such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.

What inclusion exists between the sets $E$ and $E^{\prime}$, where $E$ is the set of real numbers $x$ for which the application $t \mapsto f(t)e^{-\lambda(t)x}$ is integrable on $\mathbb{R}^+$, and $E^{\prime}$ is the set of real numbers $x$ for which the integral $\int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt$ converges?

Show that if $E$ is not empty, then $E$ is an unbounded interval of $\mathbb{R}$.

Show that if $E$ is not empty, then $Lf$ is continuous on $E$, where for $x \in E^{\prime}$, $$Lf(x) = \int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt.$$

Compare $E$ and $E^{\prime}$ in the case where $f$ is positive.