2. We define a relation $\sim$ on $S$ as follows: for every pair $(f, g)$ of $S^2$, $f \sim g$ if and only if there exist two continuous bijections $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ and $\psi : \mathbb{R} \rightarrow \mathbb{R}$, strictly increasing, which satisfy $f = \psi \circ g \circ \varphi$. a. Verify that $\sim$ is an equivalence relation on $S$ and show that each equivalence class of $\sim$ is contained in one of the sets $S_n, n \in \mathbb{N}$. b. Let $n \in \mathbb{N}^*$ and $\{u_1, \ldots, u_n\}, \{v_1, \ldots, v_n\}$ be subsets of $\mathbb{R}$ satisfying $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Verify that there exists a continuous bijection $\chi : \mathbb{R} \rightarrow \mathbb{R}$ strictly increasing such that $\chi(u_k) = v_k$ for $1 \leq k \leq n$. c. Suppose that $f$ and $g$ are in $S_*$ and that $$\lim_{x \rightarrow \pm\infty} |f(x)| = +\infty, \quad \lim_{x \rightarrow \pm\infty} |g(x)| = +\infty$$ Prove that $f \sim g$ if and only if $\sigma_f = \sigma_g$. d. Does the preceding equivalence hold for two arbitrary functions $f$ and $g$ of $S_*$?

3. We denote $C_b^0$ the space of continuous bounded functions from $\mathbb{R}$ to $\mathbb{R}$, equipped with the uniform norm: $\|f\| = \operatorname{Sup}_{x \in \mathbb{R}} |f(x)|$ for $f \in C_b^0$. a. Let $n \in \mathbb{N}^*, \{u_1, \ldots, u_n\} \subset \mathbb{R}$ and $\{v_1, \ldots, v_n\} \subset \mathbb{R}$ with $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Show that there exists a continuous map $\zeta : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ such that:

• for $s \in [0,1]$, the function $x \mapsto \zeta(s, x)$ is a strictly increasing bijection from $\mathbb{R}$ to $\mathbb{R}$,
• $\zeta(0, x) = x$ for $x \in \mathbb{R}$ and $\zeta(1, u_k) = v_k$, $1 \leq k \leq n$.

b. Prove that the equivalence classes of the restriction of $\sim$ to $S_* \cap C_b^0$ are arc-connected. c. Give an example of a continuous arc $\gamma : [0,1] \rightarrow S \cap C_b^0$ such that $\gamma(0) \in S_0$ and $\gamma(1) \in S_2$.

1. Let $n \in \mathbb{N}^*$. We denote Id the identity map of $\mathbb{R}^n$. We equip $\mathbb{R}^n$ with a norm denoted $\|\cdot\|$ and the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the associated norm, also denoted $\|\cdot\|$. For $x \in \mathbb{R}^n$ and $r \in \mathbb{R}^+$, we denote $B(x, r)$ (resp. $B(x, r]$) the open (resp. closed) ball with center $x$ and radius $r$. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ containing 0 and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ such that $f(0) = 0$ and whose differential $\varphi$ at 0 is invertible. a. We set $g = \mathrm{Id} - \varphi^{-1} \circ f$. Show that $g$ is of class $C^1$ on $\mathcal{O}$ and that there exists $\varepsilon > 0$ such that $B(0, \varepsilon) \subset \mathcal{O}$ and $\|Dg(x)\| \leq \frac{1}{2}$ for $x \in B(0, \varepsilon)$. Deduce that $f$ is injective in $B(0, \varepsilon)$. b. Let $0 < r < \varepsilon$ and let $z_0 \in B(0, r/2)$. We set $h(x) = g(x) + z_0$ for $x \in \mathcal{O}$. Show that $$h(B(0, r]) \subset B(0, r].$$ c. Show that there exists $a \in B(0, r]$ such that $f(a) = \varphi(z_0)$. d. Let $W = \varphi(B(0, r/2))$ and $V = f^{-1}(W) \cap B(0, \varepsilon)$. Show that $V$ and $W$ are open and that $f_{|V}$ is a homeomorphism from $V$ to $W$.

2. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ whose differential at $x$ is invertible for all $x \in \mathcal{O}$. Prove that the image by $f$ of an open set of $\mathcal{O}$ is an open set of $\mathbb{R}^n$.

3. For $n \geq 2$, let $O_{n-1} = \{(x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1} \mid 0 < x_1 < x_2 < \cdots < x_{n-1}\}$ and let $U_{n-1}$ be the set of $(n-1)$-tuples $(y_1, \ldots, y_{n-1}) \in \mathbb{R}^{n-1}$ such that $$0 < y_1, \quad y_i > y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is odd}, \quad y_i < y_{i+1} \text{ if } i \in \{1, \ldots, n-2\} \text{ is even}.$$ For $x \in O_{n-1}$, we define the function $\pi_x \in \mathscr{P}_n$ by $\pi_x(t) = t(x_1 - t) \cdots (x_{n-1} - t)$. We define the map $Y = (Y_1, \ldots, Y_{n-1}) : O_{n-1} \rightarrow \mathbb{R}^{n-1}$ by $$Y_i(x) = \int_0^{x_i} \pi_x(u)\, du, \quad x = (x_1, \ldots, x_{n-1}) \in O_{n-1}$$ a. Let $j \in \{1, \ldots, n-1\}$ and $x \in O_{n-1}$. Show that $$d_{x,j} : t \mapsto \int_0^t u \prod_{1 \leq \ell \leq n-1, \ell \neq j} (x_\ell - u)\, du$$ is in $\mathscr{P}_n$ and vanishes with its derivative at 0. Deduce the existence of $\chi_{x,j} \in \mathscr{P}_{n-2}$ satisfying $$\forall t \in \mathbb{R}, \quad d_{x,j}(t) = t^2 \chi_{x,j}(t).$$ b. For $x \in O_{n-1}$ and $(i,j) \in \{1, \ldots, n-1\}^2$, show the existence of $\frac{\partial Y_i}{\partial x_j}(x)$ and verify that $$\frac{\partial Y_i}{\partial x_j}(x) = d_{x,j}(x_i)$$ Deduce that $Y$ is a map of class $C^1$ on the open set $O_{n-1}$, with values in $U_{n-1}$. c. Prove that for $x \in O_{n-1}$, the set $\{\chi_{x,j} \mid j \in \{1, \ldots, n-1\}\}$ is a basis of $\mathscr{P}_{n-2}$. d. Deduce that the differential of $Y$ at point $x$ is invertible.

4. For $n \in \mathbb{N}$, a function of $\mathscr{P}_n$ is said to be monic when the coefficient of its term of degree $n$ is 1. We denote $\mathscr{P}_n^u$ the set of these functions. We denote $C_n = \operatorname{Inf}\left\{\int_0^1 |f(t)|\, dt \mid f \in \mathscr{P}_n^u\right\}$. a. Show that $C_n > 0$. b. For $n \geq 2$, prove that if $x \in O_{n-1}$ $$(x_{n-1})^{n+1} \leq \frac{1}{C_n}\left[Y_1(x) + \sum_{i=1}^{n-2} (-1)^i (Y_{i+1}(x) - Y_i(x))\right]$$ c. Verify that the map $Y$ extends continuously to the closure of $O_{n-1}$. d. Show that if $K$ is a compact subset of $\mathbb{R}^{n-1}$ contained in $U_{n-1}$, $Y^{-1}(K)$ is compact.

5. Show that $Y(O_{n-1})$ is open and closed in $U_{n-1}$ and deduce that $Y$ is surjective.

6. Show that for all $n \in \mathbb{N}$, for every function $f$ of $S_n$ satisfying $\lim_{x \rightarrow \pm\infty} |f(x)| = \pm\infty$, there exists an element $g \in \mathscr{P}_{n+1}$ such that $f \sim g$ (where $\sim$ is the relation defined in I.2).

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Show that $\varphi$ is of class $\mathcal{C}^1$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Calculate $\|\gamma(t)\|$ then justify that $\varphi'(0) = 0$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying, $$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$ Suppose that there exists a unit vector $x_0 \in F$ satisfying $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ Show that $x_0$ is an eigenvector of $u$.

Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by $$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$
For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.

For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Show that $K$ is continuous on $[0,1] \times [0,1]$.

For all $f, g \in \mathcal{E}$, we define $$(f \mid g) = \int_{-\infty}^{+\infty} f(y) g(y) \,\mathrm{d}y.$$
We define $\gamma_\lambda : \mathbf{R} \rightarrow \mathbf{R}$ by $\gamma_\lambda(y) = \exp\left(-y^2/\lambda\right)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$.
(a) Show that for all $f \in \mathcal{E}$, we have $(f \mid f) \geq 0$ with equality if and only if $f = 0$.
(b) Show that for all $x \in \mathbf{R}$, $\tau_x\left(\gamma_\lambda\right)$ belongs to $\mathcal{E}$.

In this part, $E$ denotes the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f,g \rangle = \int_0^1 f(t)g(t)\,\mathrm{d}t$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ Show that $T$ is a continuous endomorphism of $E$.

We define $\gamma_\lambda(y) = \exp(-y^2/\lambda)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$. For all $f, g \in \mathcal{E}$, $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$.
(a) Let $a > 0$. Show that there exists $c \geq 0$ such that for all $x \in \mathbf{R}$ we have $$\int_{-\infty}^{+\infty} \exp\left(-\frac{(y-x)^2}{\lambda}\right) \exp\left(-\frac{y^2}{a}\right) \mathrm{d}y = c \exp\left(-\frac{x^2}{a+\lambda}\right).$$ Hint: One may show the equality $$\frac{(y-x)^2}{\lambda} + \frac{y^2}{a} = \frac{a+\lambda}{a\lambda}\left(y - \frac{ax}{a+\lambda}\right)^2 + \frac{x^2}{a+\lambda}.$$
(b) Let $g \in \mathcal{E}$. We consider $C(g) : \mathbf{R} \rightarrow \mathbf{R}$ defined for all $x \in \mathbf{R}$ by $$C(g)(x) = \left(\tau_x(\gamma_\lambda) \mid g\right)$$ Show that $C(g) \in \mathcal{E}$.
(c) Show that $C : \mathcal{E} \rightarrow \mathcal{E}$ defines an endomorphism of $\mathcal{E}$.

Let $F$ be the vector subspace of $E$ formed of polynomial functions. For $k \in \mathbb{N}$, we denote by $p_k$ the function defined by $p_k(x) = x^k$. For all $k \in \mathbb{N}$, calculate $T(p_k)$. Deduce that $F$ is stable under $T$.

Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$: $$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.

Let $F$ be the vector subspace of $E$ formed of polynomial functions. For $k \in \mathbb{N}$, we denote by $p_k$ the function defined by $p_k(x) = x^k$. Deduce $(T(p))''$ for all $p \in F$.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
11a. Show that the set of totally real numbers is a subfield of $\mathbb { R }$. (One may use the result of question 9.)
11b. Show that the set of totally positive numbers is contained in $\mathbb { R } _ { + }$, is closed under addition and multiplication, and that the inverse of a non-zero totally positive number is totally positive.

Let $\lambda > 0$ be fixed. We use the notation $\mathcal{G}$, $\mathcal{H} = C(\mathcal{G})$, $\gamma_\lambda$, $\tau_x$, and $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$ as defined previously.
(a) Show that there exists $c_\lambda > 0$ such that for all $(x, x') \in \mathbf{R} \times \mathbf{R}$ we have $$\left(\tau_x(\gamma_\lambda) \mid \tau_{x'}(\gamma_\lambda)\right) = c_\lambda \gamma_{2\lambda}(x - x')$$ Hint: One may note that $\frac{1}{\lambda}\left((y-x)^2 + (y-x')^2\right) = \frac{2}{\lambda}\left(y - (x+x')/2\right)^2 + \frac{1}{2\lambda}(x'-x)^2$.
(b) Deduce that for all $x \in \mathbf{R}$ $$C\left(\tau_x(\gamma_\lambda)\right) = c_\lambda \tau_x(\gamma_{2\lambda})$$ and that $$\mathcal{H} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda}) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ Let $f \in E$. Calculate $T(f)(0)$ and $T(f)(1)$.

Conclude that $$\frac { E \left( N _ { n } \right) } { n } \underset { n \rightarrow + \infty } { \longrightarrow } P ( R = + \infty ) .$$ One may admit and use Cesàro's theorem: if $\left( u _ { n } \right) _ { n \in \mathbb{N}^{*} }$ is a real sequence converging to the real number $\ell$, then $$\frac { 1 } { n } \sum _ { k = 1 } ^ { n } u _ { k } \underset { n \rightarrow + \infty } { \longrightarrow } \ell .$$

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $x$ be a complex number. Show that $x$ is totally real if and only if $x ^ { 2 }$ is totally positive.