For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $x, y \in \mathbb{R}$. We set $z = \lambda x + (1-\lambda) y$. Prove that if $|y| \leq M$ then $\Psi(x) \leq \Psi_{M}(z)$. Similarly, prove that if $|x| \leq M$ then $\Psi(y) \leq \Psi_{M}(z)$.

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $\epsilon \in ]0,1[$, $f_{\epsilon} = f + \epsilon\Psi$ and $g_{\epsilon} = g + \epsilon\Psi$. Show that $$\forall x, y \in \mathbb{R}, \quad f_{\epsilon}(x)^{\lambda} g_{\epsilon}(y)^{1-\lambda} \leq h(z) + \epsilon^{\Lambda}\left(\|f\|_{\infty}^{\lambda} + \|g\|_{\infty}^{1-\lambda}\right)\left(\Psi_{M}(z)\right)^{\Lambda} + \epsilon\Psi(z)$$ where $z = \lambda x + (1-\lambda) y$. One should begin by applying the inequality from question 2, then the two preceding questions. (We recall that $f(x) = 0$ if $|x| > M$ and that $g(y) = 0$ if $|y| > M$).

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$.
Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.

For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{2}$. Show that there exists a constant $M' \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M' 4^{-j}$.

Let $s \in ]0,1[$. Show that if $f \in \Gamma^{s}(x_{0}) \cap \mathcal{C}_{0}$, then there exists a real number $c_{1} > 0$, such that for all $(j, k) \in \mathcal{I}$, we have $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s} .$$ Recall that $c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2}$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
Deduce that $\operatorname{dim}(G) \leq 2m-2$.

Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Deduce that there exist three reals $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $t \in ] 0 , t _ { 0 } ]$ and all $\theta \in [ - \pi , \pi ]$,
$$\left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \beta \left( t ^ { - 3 / 2 } \theta \right) ^ { 2 } } \quad \text { or } \quad \left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \gamma \left( t ^ { - 3 / 2 } | \theta | \right) ^ { 2 / 3 } }$$

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
$\mathbf{6}$ ▷ Prove the inequalities $$\forall k \in \mathbf{N}^* \quad \left\| X_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right) \text{ and } \left\| Y_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right).$$

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
We introduce the function $$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$
$\mathbf{7}$ ▷ Show that $$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$

Let $\mathcal { W }$ be a vector subspace of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ of dimension $d$. Prove that $$\operatorname { card } \left( \mathcal { W } \cap \mathcal { V } _ { n , 1 } \right) \leqslant 2 ^ { d }$$

We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that the real number $$C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$$ exists and belongs to the interval $[ 0,1 ]$.

We denote by $\mathcal{R}_{n}$ the set of rational functions with no pole in $\mathbb{U}$ of the form $\frac{P}{Q}$ where $P$ and $Q$ are two elements of $\mathbf{C}_{n}[X]$.
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $$\forall z \in \mathbb{U}, \quad Q(z) \neq 0$$ For $t \in [-\pi, \pi]$, we set $$f(t) = F\left(e^{it}\right) = g(t) + ih(t) \quad \text{where} \quad (g(t), h(t)) \in \mathbf{R}^{2}$$ For $u \in [-\pi, \pi]$, we define a function $f_{u}$ from $[-\pi, \pi]$ to $\mathbf{R}$ by $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = g(t)\cos(u) + h(t)\sin(u) = \operatorname{Re}\left(e^{-iu}F\left(e^{it}\right)\right) = \operatorname{Re}\left(e^{-iu}f(t)\right).$$
In this question, we fix $u \in [-\pi, \pi]$ and assume that $f_{u}$ is not constant. We also fix $y \in \mathbf{R}$. Using if necessary the expression of $f_{u}(t)$ as the real part of $e^{-iu}F\left(e^{it}\right)$ and Euler's formula for the real part, determine $S \in \mathbf{C}_{2n}[X]$ such that $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = y \Longleftrightarrow S\left(e^{it}\right) = 0.$$ Deduce that the set $f_{u}^{-1}(\{y\}) \cap [-\pi, \pi[$ is finite with cardinality bounded by $2n$.

Show that there exists a constant $C>0$ such that for all $v\in\mathcal{H}$, $$|\{g\in\Gamma \text{ such that } gv\in T\}| \leq C.$$

For all $R\in\mathbb{R}$, set $$\Gamma(R) = \{g\in\Gamma \text{ such that } d(v_0,gv_0)\leq R\}.$$ Recall that $\Gamma(R)$ is a finite set. Let $D = \sup_{v\in T} d(v_0,v)$.
Show that, for all $s\geq 0$, $$\frac{1}{C}|\Gamma(\operatorname{arcch}(s)-D)|\cdot|P_k\cap T| \leq |P_k(s)| \leq |\Gamma(\operatorname{arcch}(s)+D)|\cdot|P_k\cap T|.$$

For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:

1. for all $g\in\Gamma(n\ln(2))$, $$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
2. for all $v\in\Delta(n)$, $$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Justify that $c \leqslant 1$.

Let $x \geqslant 3$. Show that
$$\pi ( x ) \geqslant \frac { \ln ( 2 ) } { 6 } \frac { x } { \ln ( x ) }$$
One may set $n = \lfloor x / 2 \rfloor$ and use Q12.

We have
$$I _ { n } = ( - 1 ) ^ { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } y ^ { n } ( 1 - y ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x \mathrm {~d} y$$
Let $n \in \mathbb { N } ^ { * }$. Deduce that
$$\left| I _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 \sqrt { 5 } - 11 } { 2 } \right) ^ { n }$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Justify that $f_h$ is bounded below by a strictly positive real number $c_h$ (which we do not seek to determine).

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
An analogous proof to that of the previous subsection allows us to show that, for any continuous and bounded function $f$ on $\mathbb{R}$, $$E_{n,f} \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
Let $K \in \mathbb{R}_+^*$, and let $f$ be a $K$-Lipschitz function and bounded on $\mathbb{R}$. Show that $$\left|E_{n,f} - \mathbb{E}\left(f\left(n^{1/4} M_n\right)\right)\right| \leqslant \frac{2K}{n^{1/4}\sqrt{2\pi}}$$ and deduce that $$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$